2024 126 13 May

Azimuthal anisotropy in coherent $\mathbf{\rho^{0}}$ photoproduction

ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration

The first measurement of the impact-parameter dependent angular anisotropy in
the decay of coherently photoproduced $\rho^{0}$ mesons is presented. The
$\rho^{0}$ mesons are reconstructed through their decay into a pion pair. The
measured anisotropy corresponds to the amplitude of the $\cos(2\phi)$
modulation, where $\phi$ is the angle between the two vectors formed by the
sum and the difference of the transverse momenta of the pions, respectively.
The measurement was performed by the ALICE Collaboration at the LHC using data
from ultraperipheral Pb–Pb collisions at a center-of-mass energy of
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV per nucleon pair. Different impact-
parameter regions are selected by classifying the events in nuclear-breakup
classes. The amplitude of the $\cos(2\phi)$ modulation is found to increase by
about one order of magnitude from large to small impact parameters.
Theoretical calculations, which describe the measurement, explain the
$\cos(2\phi)$ anisotropy as the result of a quantum interference effect at the
femtometer scale that arises from the ambiguity as to which of the nuclei is
the source of the photon in the interaction.

##  1 Introduction

The heavy ions circulating at the Relativistic Heavy-Ion Collider (RHIC) and
Large Hadron Collider (LHC) accelerators are accompanied by a strong, Lorentz
contracted, electromagnetic field that can be described as a flux of quasi-
real photons. This flux makes it possible to study photoproduction
interactions at these facilities [1, 2, 3, 4]. Most of the experimental work
uses ultraperipheral collisions (UPCs) where the colliding ions cross paths at
impact parameters larger than the sum of their radii. Given the short range of
the strong force, UPCs allow one to separate photon-induced processes from
hadronic interactions.

One of the processes that has received great interest is the photonuclear
production of a vector meson, where the incoming photon fluctuates into a
quark–antiquark color dipole that scatters off the nucleus traveling in the
opposite direction (denoted as the target), and appears as a real vector
meson. This process can be either coherent, if the photon couples to the
nucleus as a whole, or incoherent, if it interacts with a single nucleon. The
two processes result in a different transverse momentum ($p_{\rm T}$)
spectrum; the mean $p_{\rm T}$ of the vector meson is related to the size of
the target in the impact-parameter plane and it is around 60 (500) MeV/$c$ in
the coherent (incoherent) case. In the coherent scenario, it is not known
which of the two colliding nuclei emits the photon and which acts as the
target, opening up the possibility to study, at femtometer scales, the
fundamental quantum mechanical interference between the amplitudes. This idea
was first proposed in Ref. [5], where it was noted that interference effects
should be stronger: ($i$) around midrapidity, where the magnitude of both
amplitudes is similar, and ($ii$) at small impact parameters $b$, where $b$
acts analogously to the distance between slits, in a two-slit interferometer.

Coherent vector meson photoproduction accompanied by electromagnetic
dissociation (EMD) offers the opportunity to select different impact parameter
regions in UPCs [6]. The electromagnetic field of the heavy ions is so intense
that there is a non-negligible probability that the two nuclei, besides
interacting to produce the $\rho^{0}$, also exchange photons in an independent
EMD interaction, where the excited nuclei emit neutrons at beam rapidities.
Experimentally, the emitted neutrons can be detected using two zero-degree
calorimeters (ZDCs) each of them covering the direction of one of the incoming
colliding nuclei. This allows for classifying UPCs as: (i) XnXn, where at
least one neutron is detected in each ZDC, (ii) Xn0n + 0nXn, where at least
one neutron is detected in only one of the ZDCs, and (iii) 0n0n, where no
neutron is detected in the ZDCs; for brevity, the class Xn0n + 0nXn will be
denoted as Xn0n in the following text. Since the intensity of the
electromagnetic field grows with decreasing impact parameter, the XnXn
configuration, where at least three photons are exchanged, selects a region of
relatively small impact parameters. The 0nXn and Xn0n configurations select a
broader impact-parameter range than XnXn, while 0n0n events encompass all
possible impact parameters. EMD is modeled in the RELDIS [7, 8] and
n$\mathbf{{}_{O}^{O}}$n [9] models, while the coherent production of vector
mesons accompanied by electromagnetic dissociation is studied with
n$\mathbf{{}_{O}^{O}}$n and STARlight [10]. According to
n$\mathbf{{}_{O}^{O}}$n, the median impact parameter of coherent $\rho^{0}$
photoproduction at the LHC energy changes from about 49 fm in 0n0n to about 19
fm in XnXn.

Ref. [5] proposes the suppression of coherent $\rho^{0}$ production at small
transverse momentum in UPCs as an observable to study interference effects.
This effect was measured by the STAR Collaboration in coherent $\rho^{0}$
photoproduction at a center-of-mass energy per nucleon pair of
$\sqrt{s_{\mathrm{NN}}}~{}=~{}200$ GeV [11]. The measurement was carried out
using two samples, one corresponding to XnXn and the other without any
requirement on the detection of neutrons at beam rapidities. As expected, it
was observed that the destructive interference was more pronounced in the XnXn
sample.

Recently, it was pointed out that the interference can also give rise to an
azimuthal anisotropy, since the incoming photons are linearly polarized. It
was suggested to look for this effect in the process $\gamma+\gamma\to
l^{+}+l^{-}$ where $\gamma$ and $l^{\pm}$ denote photons and leptons,
respectively [12]. The dependence of this phenomenon on the impact parameter
was studied in Ref. [13]. Shortly thereafter, this effect was measured, for
the XnXn event class, by the STAR Collaboration in Au–Au UPCs at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}200$ GeV [14].

These studies were later extended to the photoproduction of a $\rho^{0}$
vector meson, where the $\rho^{0}$ inherits the linear polarization of the
photon, giving rise to a $\cos(2\phi)$ asymmetry [15, 16]. Here, $\phi$ is the
angle between the two vectors formed by the sum and by the difference of the
transverse momenta of the pions produced in the decay
$\rho^{0}\to\pi^{+}\pi^{-}$. More recently, it was proposed to look for
$\cos(\phi)$, $\cos(3\phi)$ [17], and $\cos(4\phi)$ [18] modulations. The
first two patterns could be produced by the interference of the production of
$\rho^{0}$ with QED processes, and the last by the interference of resonant
and open production of pion pairs. It was also proposed to search for
asymmetries in the photoproduction of a J$/\psi$ vector meson [19]. The
predicted $\cos(2\phi)$ modulation was measured by the STAR Collaboration, for
$\rho^{0}$ coherent production in XnXn events, in Au–Au and U–U UPCs at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}200$ GeV and $\sqrt{s_{\mathrm{NN}}}~{}=~{}193$
GeV, respectively [20]. This asymmetry was also recently studied by the CMS
Collaboration using exclusive diffractive production of jets at the LHC [21].

The ALICE Collaboration has measured the cross section for EMD in Pb–Pb
collisions at center-of-mass energies of $\sqrt{s_{\mathrm{NN}}}~{}=~{}2.76$
TeV [22] and $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [23], where a good
agreement with the predictions from RELDIS and n$\mathbf{{}_{O}^{O}}$n was
found. The ALICE Collaboration has also measured coherent $\rho^{0}$
photoproduction in Pb–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}~{}=~{}2.76$ TeV [24]
and $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV [25], as well as in Xe–Xe UPCs at
$\sqrt{s_{\mathrm{NN}}}~{}=~{}5.44$ TeV [26]. The results were compared to
predictions from the STARlight and GDL [27] models in Ref. [24], and from
STARlight and n$\mathbf{{}_{O}^{O}}$n in Refs. [25, 26]; in general, the
tested models describe well the relative $\rho^{0}$ yields in the 0n0n, Xn0n,
and XnXn classes. These measurements demonstrate that coherent $\rho^{0}$
photoproduction accompanied by EMD is well understood at the LHC and that
neutron emission in EMD can be used to select different event classes which
are dominated by different impact parameter ranges.

In this Letter, the impact-parameter dependence of the $\cos(2\phi)$ asymmetry
is studied in Pb–Pb UPCs at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV using the
coherent photoproduction of a $\rho^{0}$ meson decaying into a pion pair. The
measurements are performed at midrapidity in the range $|y|<0.8$ and in three
different EMD classes: 0n0n, Xn0n, and XnXn.

##  2 Experimental set-up

A full description of the ALICE apparatus and its performance is given in
Refs. [28, 29]. A brief description of the sub-detectors involved in this
analysis is given hereafter. The $\rho^{0}$ meson is detected through its
decay into a pion pair at midrapidity, using the Inner Tracking System (ITS)
[30] and the Time Projection Chamber (TPC) [31] to reconstruct the pion
tracks. The V0 [32] and ALICE Diffractive (AD) [33] detectors, located at
forward rapidities, provide a veto, suppressing hadronic interactions. As
mentioned in Sec. 1, three different impact-parameter ranges are selected by
means of neutrons emitted at forward rapidities, measured by the ZDCs [22].

The ITS is composed of six cylindrical layers coaxial with the beam line.
Three different technologies are used, starting from the inner layer: pixels,
drift, and strip sensors; each technology is used in two consecutive layers.
All six layers are used for tracking, while the two innermost layers, the
Silicon Pixel Detector (SPD), are also used for triggering. The TPC is a large
cylindrical gaseous detector that surrounds the ITS. It has a central cathode
at high voltage and two readout planes at the end caps, composed of multiwire
proportional chambers. It is the main tracking detector and provides particle
identification (PID) by measuring the specific ionization energy loss. The ITS
and the TPC cover a pseudorapidity interval $|\eta|<0.9$ and the full azimuth;
they are located inside a solenoid magnet that provides a magnetic field of
$B=0.5$ T.

The V0 is composed of two scintillator arrays, V0A and V0C, installed on both
sides of the nominal interaction point (IP). They cover the pseudorapidity
ranges $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$, respectively. The AD consists of
two scintillator stations, ADA and ADC, located along the beam line at $+16$ m
and $-19$ m from the IP and covering the pseudorapidity ranges $4.8<\eta<6.3$
and $-7.0<\eta<-4.9$, respectively.

There are two ZDC detectors for neutrons, ZNA and ZNC, located at $\pm 112.5$
m from the IP along the beam line. They detect neutrons with $|\eta|>8.8$,
with an energy resolution good enough to be sensitive to the emission of a
single neutron. Neutron signals are discriminated with a threshold
corresponding to an energy deposition of $\sim$ 1 TeV, which is about three
standard deviations below the expected signal from a 2.51 TeV neutron. The ZNs
also determine the arrival time of the particles, allowing for the rejection
of beam–gas interactions involving charge circulating outside the nominal LHC
bunch positions.

The analyzed data were recorded by ALICE in 2015, when the LHC provided Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}~{}=~{}5.02$ TeV, using a dedicated UPC
trigger. This trigger exploits five different signals: four of them veto any
activity on either side of the AD or V0 detector within the time window for
nominal beam–beam interactions, to suppress hadronic collisions. The fifth
signal is a topological trigger that selects events that have at least two
track segments, defined as in Ref. [25], in the SPD, with an opening angle in
azimuth greater than 153 degrees. This topology was chosen since the
coherently produced $\rho^{0}$ has a very small transverse momentum and hence
the tracks of the pions are almost back-to-back in azimuth. The integrated
luminosity of the sample, determined using the V0 detectors as explained in
Ref. [25], is about 485 mb-1.

##  3 Analysis procedure

###  3.1 Track and event selection

Tracks were required to have a distance of closest approach to the event
primary vertex smaller than
$0.0182~{}+~{}0.0350/(p_{\text{T}}^{\text{trk}})^{1.01}$ cm in the transverse
plane and smaller than 2 cm in the longitudinal direction, where
$p_{\text{T}}^{\text{trk}}$ is the transverse momentum, in GeV/$c$, associated
to the track. Tracks were also required to have more than 50 associated hits
in the TPC, to be reconstructed in both ITS and TPC, and to match the track
segments in the SPD that fired the trigger.

The events with good tracks were required to fulfill additional selections:
(i) have exactly two tracks of opposite sign, (ii) have no offline
reconstructed signal in neither the V0 nor AD detectors, and (iii) fulfill the
pion selection $\text{n}_{\sigma 1}^{2}+\text{n}_{\sigma 2}^{2}<5^{2}$, where
$\text{n}_{\sigma 1}$ ($\text{n}_{\sigma 2}$) is the difference, in units of
the TPC ionization energy loss resolution, between the measured energy loss
for track 1 (track 2) and the expected value for a pion with the same
momentum.

Kinematic selections were also applied: (i) the pion pair rapidity lies in the
range $|y|<0.8$ to avoid acceptance edge effects, (ii) the invariant mass of
the pion pair is inside the range
$0.6~{}\text{Ge\kern-1.00006ptV/$c^{2}$}<\text{$m_{\pi\pi}$}~{}<0.95\text{
Ge\kern-1.00006ptV/$c^{2}$}$, and (iii) the transverse momentum of the
$\rho^{0}$ candidate is less than 0.1 GeV/$c$ to select coherent processes
with high purity. With such a selection, the contamination from incoherent
events is found to be lower than 4% [25]. More details about event and track
selections can be found in Ref. [25].

The data were divided in three independent classes, based on the detection of
neutrons at forward rapidity. Events with neutron emission were selected by
requiring a signal in ZNA and/or ZNC. The signal time was required to lie
within 2 ns from the nominal collision time. As explained in Sec. 1, these
classes (0n0n, Xn0n, XnXn) can be used to select different impact-parameter
ranges for ultraperipheral collisions. For each neutron class, the data were
arranged in seven $\phi$ intervals of equal size, with $\phi$ defined as in
Sec. 3.2. The efficiency correction (Sec. 3.3) and signal extraction (Sec.
3.4) were then performed separately for each neutron class and $\phi$
interval.

###  3.2 Asymmetry angle definition

The anisotropy described in Sec. 1 is predicted to be strongly visible as a
function of a variable called $\phi$, defined using the momenta of the pions
($\vec{\pi}_{1,2}$) into which the $\rho^{0}$ decays. The $\phi$ angle can be
defined in two different ways, which will be referred to as average and
charge, respectively. In both cases $\phi$ is defined as the angle between the
transverse components of $\vec{p}_{+}$ and $\vec{p}_{-}$, where
$\vec{p}_{\pm}=\vec{\pi}_{1}\pm\vec{\pi}_{2}$. Using the average definition,
$\vec{\pi}_{1,2}$ are randomly associated to the positive or to the negative
track. Using the charge definition, $\vec{\pi}_{1}$ and $\vec{\pi}_{2}$ are
the momenta of the positive and of the negative track, respectively. The
average definition is helpful since, by construction, it does not allow for a
$\cos(\phi)$ component. The two definitions are equivalent in terms of the
predicted cos(2$\phi$) component. The average definition was chosen as the
default one, while the charge definition was used in the evaluation of the
systematic uncertainties. In both cases, the $\phi$ angle was initially
computed within the range of $-\pi$ to $\pi$. Then, since cos($\phi$) and
cos(2$\phi$) are even functions, the resulting values were remapped between 0
and $\pi$ by flipping the sign of negative values. This procedure improves the
stability of signal extraction when fitting the invariant mass spectra in each
$\phi$ interval, by doubling the size of the available sample.

###  3.3 Corrections

The correction for the acceptance and efficiency of the detector for the
reconstruction and selection of the pion tracks, $\rm{Acc\times\epsilon}$, was
estimated as a function of the pion pair invariant mass, using the STARlight
[10] Monte Carlo (MC) generator and a realistic description of the ALICE
apparatus. This MC production describes accurately the raw data on the vector
meson kinematics, with the exception of the transverse momentum distribution
[24]. In order to improve the agreement of the MC with data, a re-weighting
procedure, described in the following, was applied to the generated
$p^{2}_{\rm T}$ spectrum. The first step of the procedure is to fit the
inclusive pion pair $p^{2}_{\rm T}$ distribution of the generated MC. For
sufficiently high transverse momentum, $p^{2}_{\rm T}$ can be approximated
with the Mandelstam variable $t$, while for very low $p^{2}_{\rm T}$ the
contribution from the transverse momentum of the photon plays an important
role and hence the approximation is no longer valid. Having this in mind, the
MC spectrum was fitted for $p^{2}_{\rm T}>(0.01)^{2}$ $($GeV/$c)^{2}$ using
the function

$\frac{\text{d}N}{\text{d$p^{2}_{\rm T}$}}=c\mid F(\mid
t\mid,~{}a_{\text{Pb}},~{}R_{\text{Pb}})\mid^{2},$ (1)

where $c$ is a normalization constant and $F(\mid t\mid)$ is the form factor
of the lead nucleus, obtained as a numerical approximation of the Fourier
transform of a Wood–Saxon function [34, 35], with fit parameters $R_{\rm{Pb}}$
and $a_{\rm{Pb}}$. The weights were then computed using

$\text{w($p_{\rm T}$)}=\frac{\mid F(\mid
t\mid,~{}a_{\text{Pb}},~{}R_{\text{X}})\mid^{2}}{\mid F(\mid
t\mid,~{}a_{\text{Pb}},~{}R_{\text{Pb}})\mid^{2}},$ (2)

where $a_{\text{Pb}}$ is fixed to the fit result and $R_{\text{X}}$ is chosen
in such a way that, after applying the weights to each event of a given
generated $p_{\rm T}$, the reconstructed $p^{2}_{\rm T}$ spectrum in the MC
best reproduces the one in the data. This is achieved by minimizing the bin-
by-bin difference between the $p_{\rm T}$ distributions of data and
reconstructed MC as a function of $R_{\text{X}}$, using a $\chi^{2}$-like
variable, in the region $p^{2}_{\rm T}>(0.01)^{2}$ $($GeV/$c)^{2}$, where the
model used for the reweighting is valid. It was verified that the best-fit
values of $R_{\text{X}}$ computed for different $\phi$ ranges were all
compatible among themselves, hence the weights were obtained utilizing the
full data set.

The $\rm{Acc\times\epsilon}$ correction was obtained using the re-weighted
STARlight MC simulations by computing the ratio of reconstructed to generated
number of pion pairs in each invariant mass and $\phi$ interval, after
applying the weights discussed above at the generation level. The number of
pion pairs found in data, for each invariant mass and $\phi$ interval, was
then divided by $\rm{Acc\times\epsilon}$, to obtain the corrected mass
spectra. This was done for each neutron class and $\phi$ range. The integrated
$\rm{Acc\times\epsilon}$ was found to slightly increase as a function of the
invariant mass, ranging from $\sim 10.5\%$ at $m_{\pi\pi}=0$.6 GeV/$c^{2}$ to
$\sim 14\%$ at $m_{\pi\pi}=0$.95 GeV/$c^{2}$.

###  3.4 Signal extraction

The corrected invariant mass spectra, in each neutron class and in each $\phi$
interval, were fitted with:

$\frac{\text{d}N}{\text{d}m_{\pi\pi}}=P(m_{\pi\pi})+n_{\mu\mu}\>M(m_{\pi\pi}),$
(3)

where $m_{\pi\pi}$ is the pion pair invariant mass, $P(m_{\pi\pi})$ is the
function used to describe the pion pair spectrum, $M(m_{\pi\pi})$ is the
shape, estimated with a dedicated MC based on the STARlight generator, of the
background originating from muons produced in the
$\gamma\gamma\rightarrow\mu^{+}\mu^{-}$ process and misidentified as pions,
and $n_{\mu\mu}$ is a normalization constant for said background. Two
different parameterizations were used for the pion spectrum. The first
parameterization uses a modified Söding model [36]:

$P(m_{\pi\pi})=\mid A\cdot BW_{\rho}+B\mid^{2},$ (4)

where $BW_{\rho}$ is the relativistic Breit–Wigner shape that describes the
$\rho^{0}$, $A$ is its amplitude, and $B$ is the amplitude of the continuum
pion pair production. The relativistic Breit–Wigner function describing the
$\rho^{0}$ resonance is:

$BW_{\rho}=\frac{\sqrt{m_{\pi\pi}\,m_{\rho}\,\Gamma_{\rho}(m_{\pi\pi})}}{m_{\pi\pi}^{2}-m_{\rho}^{2}+i\,m_{\rho}\,\Gamma_{\rho}(m_{\pi\pi})},$
(5)

where $m_{\rho}$ is the $\rho^{0}$ pole mass and

$\Gamma_{\rho}(m_{\pi\pi})=\Gamma(m_{\rho})\,\frac{m_{\rho}}{m_{\pi\pi}}\,\bigg{(}\frac{m_{\pi\pi}^{2}-4m_{\pi}^{2}}{m_{\rho}^{2}-4m_{\pi}^{2}}\bigg{)}^{3/2},$
(6)

where $\Gamma$($m_{\rho}$) is the $\rho^{0}$ pole width. Since $BW_{\rho}$ is
complex, an interference term between the $\rho^{0}$ and the continuum arises
from the square module in Eq. (4). The second parameterization uses the model
by Ross and Stodolsky [37]:

$P(m_{\pi\pi})=f\mid
BW_{\rho}\mid^{2}\big{(}\frac{m_{\rho}}{m_{\pi\pi}}\big{)}^{k},$ (7)

with fit parameters $f$ and $k$.

The fits were performed by fixing $m_{\rho}$ and $\Gamma(m_{\rho})$ to the
central values reported for a $\rho^{0}$ formed in a photoproduction reaction,
namely, $m_{\rho}$ = 769.2 MeV/$c^{2}$ and $\Gamma(m_{\rho})$ = 151.5
MeV/$c^{2}$ [38]. As discussed in Ref. [25], it was verified that the
extracted $\rho^{0}$ yield does not vary significantly when the fit function
is modified to include a contribution from $\omega$ decays.

In order to assess the stability of the signal extraction, the latter was
carried out, for each $\phi$ interval, with 48 different strategies, obtained
by: employing 12 different combinations of bin size and fit range (the widest
tested range being 0.6-0.95 GeV/$c^{2}$ and the narrowest 0.65-0.9
GeV/$c^{2}$), using the Söding or the Ross-Stodolsky model to describe the
pion pair mass spectrum, and using $n_{\mu\mu}$ as a free fit parameter or
fixing it to zero. An example of the mass fits, for a specific $\phi$ interval
and for the 0n0n and XnXn classes, is shown in Fig. 1. After the fit, the
$\rho^{0}$ yield is obtained by integrating the signal function,
$|A\>BW_{\rho}|^{2}$ for the Söding model and $f|BW_{\rho}|^{2}$ for the Ross-
Stodolsky model, in the mass range
$0.6<\text{$m_{\pi\pi}$}\enspace(\text{Ge\kern-1.00006ptV/$c^{2}$})<0.95$.
Such a range was chosen to be consistent with the STAR measurement [20] and
with available theory calculations (see Sec. 4 for details).

Figure 1: Invariant-mass distribution of pion pairs, with superimposed Söding
(solid line) and Ross-Stodolsky (dotted line) fits, for the range
$0<\phi<\pi/7$ in the 0n0n (left) and XnXn (right) neutron classes. The
different components of the pion-pair production amplitude in the Söding model
are shown: the Breit–Wigner shape that describes the $\rho^{0}$ (finer dotted
line), the continuum process (dash-dotted line), and the interference between
the $\rho^{0}$ and the continuum (dash-dot-dot-dot line). The Breit–Wigner
extracted from the Ross-Stodolsky model (finest dotted line) is also shown. In
this example, the background contribution from misidentified muons is fixed to
zero in the fit.

###  3.5 Asymmetry extraction

The extraction of the amplitude of the modulation is affected by the migration
of events between neutron classes, due to ZDC detector efficiency and pile-up
effects, as discussed in Ref. [25]. To take this into account, a simultaneous
fit to the measured $\rho^{0}$ yield as a function of $\phi$ in all three
experimental classes (0n0n, Xn0n, XnXn) was performed, using the following
expression:

$\begin{pmatrix}n_{\rho\>\text{0n0n}}(\phi)\\\ n_{\rho\>\text{Xn0n}}(\phi)\\\
n_{\rho\>\text{XnXn}}(\phi)\end{pmatrix}=\begin{pmatrix}1\\\ 1\\\
1\end{pmatrix}+\begin{pmatrix}w_{\text{ 0n0n $\rightarrow$ 0n0n}}&w_{\text{
Xn0n $\rightarrow$ 0n0n}}&w_{\text{ XnXn $\rightarrow$ 0n0n}}\\\ w_{\text{
0n0n $\rightarrow$ Xn0n}}&w_{\text{ Xn0n $\rightarrow$ Xn0n}}&w_{\text{ XnXn
$\rightarrow$ Xn0n}}\\\ w_{\text{ 0n0n $\rightarrow$ XnXn}}&w_{\text{ Xn0n
$\rightarrow$ XnXn}}&w_{\text{ XnXn $\rightarrow$ XnXn}}\\\
\end{pmatrix}\begin{pmatrix}a_{2\>\text{0n0n}}\\\ a_{2\>\text{Xn0n}}\\\
a_{2\>\text{XnXn}}\\\ \end{pmatrix}\,\cos(2\phi),$ (8)

where $n_{\rho\>\text{0n0n}}$ is the normalized $\rho^{0}$ yield in a given
$\phi$ range for the experimental 0n0n class, and similarly for other classes,
and the fitting parameters $a_{2\>\text{0n0n}}$, $a_{2\>\text{Xn0n}}$, and
$a_{2\>\text{XnXn}}$ are the amplitudes of the $\cos(2\phi)$ modulation in the
corresponding three physical classes. The coefficients $w_{\text{ Y
$\rightarrow$ Z}}$ represent the contribution of the physical neutron class Y
to the yield in the experimental neutron class Z, computed using the measured
cross-section ratios and migration probabilities as determined in Ref. [25].
The constant term is fixed to unity by normalization. An example of this
simultaneous fit is shown in Fig. 2.

The central value and the statistical uncertainty of the cos(2$\phi$)
modulation amplitude were determined, for each neutron emission class, by
averaging the results obtained with the 48 fit configurations described in
Sec. 3.4.

Figure 2: Example of a simultaneous fit to the $\rho^{0}$ yield as a function
of $\phi$, used to extract the amplitude of the $\cos(2\phi)$ modulation in
all neutron classes. The contribution of each physical class to the yield in
all experimental classes is shown.

###  3.6 Systematic uncertainties

The systematic uncertainty related to the signal extraction includes the
effect on the extracted cos(2$\phi$) amplitudes of variations in the strategy
for fitting the invariant mass spectra. These include: binning, range,
modeling of the $\rho^{0}$ signal, and treatment of background. The
uncertainty was obtained as the standard deviation of the distribution of the
fitted amplitudes over the 48 trials mentioned in Sec. 3.4. This uncertainty
is $12\%$ for 0n0n, $9\%$ for Xn0n, and $13\%$ for XnXn.

An additional systematic uncertainty, related to the definition of the $\phi$
angle, was estimated by testing two variations in the analysis strategy. In
the first variation, $\phi$ is computed according to the charge definition
mentioned in Sec. 3.2. In this case, the yield distribution can have a
sizeable $\cos(\phi)$ component [17], which is added to the fit function of
Eq. (8), with its amplitude as an additional free parameter. In the second
variation, the average definition of $\phi$ is used, as in the default
strategy, but a $\cos(\phi)$ component is nevertheless added to the fit
function. The systematic uncertainty was evaluated in each class as the
largest difference between the result obtained with the default setting and
that obtained with the two strategies presented in this paragraph. It amounts
to $3.6\%$ for 0n0n, $5.6\%$ for Xn0n, and $3.3\%$ for XnXn.

As a consistency check for the $\rm{Acc\times\epsilon}$ correction, the
analysis was repeated in several rapidity sub-ranges, each containing
approximately half the total number of reconstructed $\rho^{0}$ candidates. In
each neutron emission class, the amplitudes extracted in sub-ranges were all
found to be compatible, within one standard deviation, with each other and
with the amplitude extracted in the full rapidity range. The uncertainty on
the $\rm{Acc\times\epsilon}$ correction arises then mainly from the re-
weighting procedure described in Sec. 3.3. It was obtained by using the two
values of $R_{\text{X}}$ for which the $\chi^{2}$ increases by one unit with
respect to the minimum, instead of the $R_{\text{X}}$ value that minimizes the
$\chi^{2}$. The systematic uncertainty is estimated in each class as the
largest difference between the results obtained with the original and with the
modified sets of weights. It amounts to $2.9\%$ for 0n0n, $0.8\%$ for Xn0n,
and $0.9\%$ for XnXn.

The systematic uncertainties related to the migration of events across neutron
classes are evaluated by propagating the uncertainties of the ZN pile-up
probability (9%) and efficiency (1%), all taken from Ref [25], to the
extraction of the cos(2$\phi$) amplitude. The resulting uncertainty from pile-
up is $0.1\%$, $2.3\%$, and $0.9\%$, respectively, for the 0n0n, Xn0n, and
XnXn classes. The uncertainty from the ZN efficiency is $0.7\%$, $0.1\%$, and
$0.1\%$, respectively, for the 0n0n, Xn0n, and XnXn classes.

The contributions to the systematic uncertainty on the amplitude of the
$\cos(2\phi$) modulation discussed in the previous paragraphs are listed in
Table LABEL:Table:syst. The total uncertainty is obtained as the quadratic sum
of all the contributions. It amounts to $12.6\%$ for 0n0n, $11\%$ for Xn0n,
and $13.3\%$ for XnXn, and is dominated by the signal extraction for all event
classes.

Table 1: Summary of the relative systematic uncertainties for the measured amplitude of the $\cos(2\phi)$ modulation. Source | Uncertainty ($\%$)
---|---
0n0n | Xn0n + 0nXn | XnXn
Signal extraction | 12 | 9.1 | 13
$\phi$ definition | 3.6 | 5.7 | 3.3
$\rm{Acc\times\epsilon}$ | 2.9 | 0.8 | 0.9
ZN pile-up | 0.1 | 2.3 | 0.9
ZN efficiency | 0.7 | 0.1 | 0.1
Total | 12.6 | 11.0 | 13.3

##  4 Results

Figure 3 shows the extracted amplitude of the $\cos(2\phi)$ modulation as a
function of the neutron class; the numerical values are reported in Table
LABEL:Table:results, along with the n$\mathbf{{}_{O}^{O}}$n MC estimates of
the median impact parameter of the collision for each neutron class. Similar
values for the median impact parameters are found using the analytical model
presented in Ref. [39]; the values for the XnXn case are also similar to those
reported in Ref. [6]. The table also reports the amplitudes predicted by two
models described below. The measured anisotropy shows a clear trend with the
impact parameter, with a significant increase, by one order of magnitude, from
0n0n to XnXn.

The results are compared with the model by H. Xing et al. [15]. In this model,
the quasi-real photon exchanged by the nuclei is treated as a color
quark–antiquark dipole, that recombines to produce a $\rho^{0}$ after
scattering off the color-glass-condensate state [40] inside the nuclei. As
discussed in Sec. 1, the $\cos(2\phi)$ anisotropy in the model emerges from
the presence of two elements: the first is that the photon is linearly
polarized along the impact parameter and this polarization is transferred to
the produced vector meson; the second is that there is an interference between
the two amplitudes that contribute to the cross section of the vector meson
photoproduction process. The interference effect increases as the impact
parameter decreases, producing a larger anisotropy for small impact
parameters. The uncertainty of the model mostly comes from the probability of
emitting a neutron from the scattered nucleus at a given impact parameter,
where the latter has been estimated using three different parametrizations
from Refs. [41, 42, 43]. The model prediction is compatible with data for all
neutron classes. The measured amplitudes are also compared to the model of W.
Zhao et al. [44], which is based on the same formalism as the model by H. Xing
et al. [15] with two main differences: (i) the interaction of the
quark–antiquark dipole with the target is implemented by computing the
corresponding Wilson lines, and (ii) the color-charge density used to obtain
the Wilson lines is varied event-by-event to represent the different possible
color configurations of the target. The quoted uncertainty in the model
originates from the statistical precision related to the finite number of
sampled configurations. This model, which predicts a milder variation of the
modulation amplitude with the neutron class as compared to Xing et al., also
gives a reasonable description of the data, with the possible exception of the
0n0n class.

For the XnXn class, the amplitude measured by ALICE is also compared with the
ones [20] measured by the STAR Collaboration for Au–Au and U–U collisions at
the center-of-mass energies of $\sqrt{s_{\text{NN}}}$ = 200 GeV and
$\sqrt{s_{\text{NN}}}$ = 193 GeV, respectively. It is found to be compatible
with both. This is consistent with the models, which, for the XnXn selection,
predict the cos(2$\phi$) modulation amplitude to vary with the colliding
nuclei and the center-of-mass energy by less than the current experimental
uncertainties.

Table 2: Amplitudes of the $\cos(2\phi)$ modulation of the $\rho^{0}$ yield as a function of $\phi$ in all neutron classes, with statistical and systematic uncertainties. An estimate of the median impact parameter of the collision in each neutron class, obtained with the n$\mathbf{{}_{O}^{O}}$n MC, is also reported, as well as the predictions by the H. Xing et al. [15] and W. Zhao et al. [44] models. Neutron class | median b (n$\mathbf{{}_{O}^{O}}$n) | amplitude | stat. | syst. | H. Xing et al. | W. Zhao et al.
---|---|---|---|---|---|---
0n0n | 49.0 fm | 0.028 | 0.011 | 0.003 | 0.015 – 0.031 | 0.042 – 0.044
Xn0n | 22.5 fm | 0.14 | 0.04 | 0.016 | 0.14 – 0.19 | 0.136 – 0.138
XnXn | 18.2 fm | 0.25 | 0.06 | 0.03 | 0.26 – 0.29 | 0.200 – 0.214
Figure 3: Amplitudes of the $\cos(2\phi)$ modulation of the $\rho^{0}$ yield
in Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV in all neutron classes.
The results are compared with the Xing et al. [15] and W. Zhao et al. [44]
model predictions and, for the XnXn class, with the STAR results [20] in Au–Au
and U–U collisions at RHIC. For all the experimental data points, statistical
uncertainties are represented with a bar and systematic uncertainties with a
box.

##  5 Summary

The first measurement of the impact-parameter dependent angular anisotropy in
the pion-pair decay of coherently photoproduced $\rho^{0}$ mesons from Pb–Pb
ultraperipheral collisions at a center-of-mass energy of
$\sqrt{s_{\text{NN}}}$ = 5.02 TeV, performed with the ALICE detector, has been
presented. The anisotropy is quantified via the distribution of the azimuthal
angle $\phi$, defined in Sec. 3.2. The impact parameter is estimated
considering neutron emission at forward rapidity. A significant, impact-
parameter dependent, $\cos(2\phi)$ modulation is observed, with the amplitude
of the modulation increasing by about one order of magnitude from the 0n0n (no
neutrons emitted, large impact parameter) to the XnXn (neutrons emitted by
both colliding nuclei, relatively small impact parameter) class. This trend is
reproduced by the theoretical models [15, 44]. The result for the XnXn class
is compatible with similar measurements by the STAR Collaboration.

The coherent photonuclear production of a vector meson in UPCs can be seen as
a double-slit experiment [45], where the interference occurs between the
amplitudes for quasi-real photon emission by either of the two colliding
nuclei. The unambiguous observation of this interference through the
measurement of the $\cos(2\phi)$ anisotropy of the $\rho^{0}$ yield is a proof
of the validity of quantum mechanics at femtometer scales. This is the first
measurement of this effect in terms of the impact-parameter dependence. The
larger data samples expected from the LHC Run 3 and Run 4 will enable a
detailed characterization of the quantum interference effects.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Bulgarian Ministry of Education and Science, within the National Roadmap for Research Infrastructures 2020-2027 (object CERN), Bulgaria; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l’Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; National Research and Innovation Agency - BRIN, Indonesia; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Pontificia Universidad Católica del Perú, Peru; Ministry of Science and Higher Education, National Science Centre and WUT ID-UB, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics, Ministry of Research and Innovation and Institute of Atomic Physics and Universitatea Nationala de Stiinta si Tehnologie Politehnica Bucuresti, Romania; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSTDA) and National Science, Research and Innovation Fund (NSRF via PMU-B B05F650021), Thailand; Turkish Energy, Nuclear and Mineral Research Agency (TENMAK), Turkey; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America. In addition, individual groups or members have received support from: Czech Science Foundation (grant no. 23-07499S), Czech Republic; European Research Council (grant no. 950692), European Union; ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data and Quantum Computing, European Union - NextGenerationEU; Academy of Finland (Center of Excellence in Quark Matter) (grant nos. 346327, 346328), Finland.

## References

  * [1] G. Baur, K. Hencken, D. Trautmann, S. Sadovsky, and Y. Kharlov, “Coherent $\gamma\gamma$ and $\gamma$A interactions in very peripheral collisions at relativistic ion colliders”, Phys.Rep. 364 (2002) 359–450, arXiv:hep-ph/0112211 [hep-ph].
  * [2] A. J. Baltz et al., “The Physics of Ultraperipheral Collisions at the LHC”, Phys. Rep. 458 (2008) 1–171, arXiv:0706.3356 [nucl-ex].
  * [3] J. G. Contreras and J. D. Tapia Takaki, “Ultra-peripheral heavy-ion collisions at the LHC”, Int. J. Mod. Phys. A30 (2015) 1542012.
  * [4] S. R. Klein and H. Mäntysaari, “Imaging the nucleus with high-energy photons”, Nature Rev. Phys. 1 (2019) 662–674, arXiv:1910.10858 [hep-ex].
  * [5] S. R. Klein and J. Nystrand, “Interference in exclusive vector meson production in heavy ion collisions”, Phys. Rev. Lett. 84 (2000) 2330–2333, arXiv:hep-ph/9909237.
  * [6] A. J. Baltz, S. R. Klein, and J. Nystrand, “Coherent vector meson photoproduction with nuclear breakup in relativistic heavy ion collisions”, Phys. Rev. Lett. 89 (2002) 012301, arXiv:nucl-th/0205031.
  * [7] I. A. Pshenichnov, J. P. Bondorf, I. N. Mishustin, A. Ventura, and S. Masetti, “Mutual heavy ion dissociation in peripheral collisions at ultrarelativistic energies”, Phys. Rev. C 64 (2001) 024903, arXiv:nucl-th/0101035.
  * [8] I. A. Pshenichnov, “Electromagnetic excitation and fragmentation of ultrarelativistic nuclei”, Phys. Part. Nucl. 42 (2011) 215–250.
  * [9] M. Broz, J. G. Contreras, and J. D. Tapia Takaki, “A generator of forward neutrons for ultra-peripheral collisions: n$\mathbf{{}_{O}^{O}}$n”, Comput. Phys. Commun. (2020) 107181, arXiv:1908.08263 [nucl-th].
  * [10] S. R. Klein, J. Nystrand, J. Seger, Y. Gorbunov, and J. Butterworth, “STARlight: A Monte Carlo simulation program for ultra-peripheral collisions of relativistic ions”, Comput. Phys. Commun. 212 (2017) 258–268, arXiv:1607.03838 [hep-ph].
  * [11] STAR Collaboration, B. I. Abelev et al., “Observation of Two-source Interference in the Photoproduction Reaction Au Au —$>$ Au Au $\rho^{0}$”, Phys. Rev. Lett. 102 (2009) 112301, arXiv:0812.1063 [nucl-ex].
  * [12] C. Li, J. Zhou, and Y.-J. Zhou, “Probing the linear polarization of photons in ultraperipheral heavy ion collisions”, Phys. Lett. B 795 (2019) 576–580, arXiv:1903.10084 [hep-ph].
  * [13] C. Li, J. Zhou, and Y.-J. Zhou, “Impact parameter dependence of the azimuthal asymmetry in lepton pair production in heavy ion collisions”, Phys. Rev. D 101 (2020) 034015, arXiv:1911.00237 [hep-ph].
  * [14] STAR Collaboration, J. Adam et al., “Measurement of $e^{+}e^{-}$ Momentum and Angular Distributions from Linearly Polarized Photon Collisions”, Phys. Rev. Lett. 127 (2021) 052302, arXiv:1910.12400 [nucl-ex].
  * [15] H. Xing, C. Zhang, J. Zhou, and Y.-J. Zhou, “The cos 2$\phi$ azimuthal asymmetry in $\rho^{0}$ meson production in ultraperipheral heavy ion collisions”, JHEP 10 (2020) 064, arXiv:2006.06206 [hep-ph].
  * [16] W. Zha, J. D. Brandenburg, L. Ruan, Z. Tang, and Z. Xu, “Exploring the double-slit interference with linearly polarized photons”, Phys. Rev. D 103 (2021) 033007, arXiv:2006.12099 [hep-ph].
  * [17] Y. Hagiwara, C. Zhang, J. Zhou, and Y.-J. Zhou, “Coulomb nuclear interference effect in dipion production in ultraperipheral heavy ion collisions”, Phys. Rev. D 103 (2021) 074013, arXiv:2011.13151 [hep-ph].
  * [18] Y. Hagiwara, C. Zhang, J. Zhou, and Y.-J. Zhou, “Probing the gluon tomography in photoproduction of dipion”, Phys. Rev. D 104 (2021) 094021, arXiv:2106.13466 [hep-ph].
  * [19] J. D. Brandenburg, Z. Xu, W. Zha, C. Zhang, J. Zhou, and Y. Zhou, “Exploring gluon tomography with polarization dependent diffractive J/$\psi$ production”, Phys. Rev. D 106 (2022) 074008, arXiv:2207.02478 [hep-ph].
  * [20] STAR Collaboration, M. Abdallah et al., “Tomography of ultrarelativistic nuclei with polarized photon-gluon collisions”, Sci. Adv. 9 (2023) eabq3903, arXiv:2204.01625 [nucl-ex].
  * [21] CMS Collaboration, A. Tumasyan et al., “Azimuthal Correlations within Exclusive Dijets with Large Momentum Transfer in Photon-Lead Collisions”, Phys. Rev. Lett. 131 (2023) 051901, arXiv:2205.00045 [nucl-ex].
  * [22] ALICE Collaboration, B. Abelev et al., “Measurement of the Cross Section for Electromagnetic Dissociation with Neutron Emission in Pb–Pb Collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV”, Phys. Rev. Lett. 109 (2012) 252302, arXiv:1203.2436 [nucl-ex].
  * [23] ALICE Collaboration, S. Acharya et al., “Neutron emission in ultraperipheral Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV”, Phys. Rev. C 107 (2023) 064902, arXiv:2209.04250 [nucl-ex].
  * [24] ALICE Collaboration, J. Adam et al., “Coherent $\rho^{0}$ photoproduction in ultra-peripheral Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV”, JHEP 09 (2015) 095, arXiv:1503.09177 [nucl-ex].
  * [25] ALICE Collaboration, S. Acharya et al., “Coherent photoproduction of $\rho^{0}$ vector mesons in ultra-peripheral Pb–Pb collisions at $\sqrt{{\mathrm{s}}_{\mathrm{NN}}}$ = 5.02 TeV”, JHEP 06 (2020) 035, arXiv:2002.10897 [nucl-ex].
  * [26] ALICE Collaboration, S. Acharya et al., “First measurement of coherent $\rho^{0}$ photoproduction in ultra-peripheral Xe–Xe collisions at $\sqrt{s_{\mathrm{NN}}}=5.44$ TeV”, Phys. Lett. B 820 (2021) 136481, arXiv:2101.02581 [nucl-ex].
  * [27] V. Rebyakova, M. Strikman, and M. Zhalov, “Coherent $\rho$ and J/$\psi$ photoproduction in ultraperipheral processes with electromagnetic dissociation of heavy ions at RHIC and LHC”, Phys. Lett. B 710 (2012) 647–653, arXiv:1109.0737 [hep-ph].
  * [28] ALICE Collaboration, K. Aamodt et al., “The ALICE experiment at the CERN LHC”, JINST 3 (2008) S08002.
  * [29] ALICE Collaboration, B. Abelev et al., “Performance of the ALICE Experiment at the CERN LHC”, Int. J. Mod. Phys. A29 (2014) 1430044, arXiv:1402.4476 [nucl-ex].
  * [30] ALICE Collaboration, K. Aamodt et al., “Alignment of the ALICE Inner Tracking System with cosmic-ray tracks”, JINST 5 (2010) P03003, arXiv:1001.0502 [physics.ins-det].
  * [31] J. Alme et al., “The ALICE TPC, a large 3-dimensional tracking device with fast readout for ultra-high multiplicity events”, Nucl. Instrum. Meth. A622 (2010) 316–367, arXiv:1001.1950 [physics.ins-det].
  * [32] ALICE Collaboration, E. Abbas et al., “Performance of the ALICE VZERO system”, JINST 8 (2013) P10016, arXiv:1306.3130 [nucl-ex].
  * [33] LHC Forward Physics Working Group Collaboration, K. Akiba et al., “LHC Forward Physics”, J. Phys. G Nucl. Part 43 (2016) 110201, arXiv:1611.05079 [hep-ph].
  * [34] S. R. Klein and J. Nystrand, “Exclusive vector meson production in relativistic heavy ion collisions”, Phys. Rev. C60 (1999) 014903, arXiv:hep-ph/9902259 [hep-ph].
  * [35] L. C. Maximon and R. A. Schrack, “The form factor of the Fermi model spatial distribution”, J. Res. Natl. Bur. Stand. B 70 (1966) 85–94.
  * [36] A. S. Krass, “Interference Effects in Photoproduction of ${\varrho}^{0}$ Mesons”, Phys. Rev. 159 (1967) 1496–1500.
  * [37] M. Ross and L. Stodolsky, “Photon Dissociation Model for Vector-Meson Photoproduction”, Phys. Rev. 149 (1966) 1172–1181.
  * [38] Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics”, PTEP 2022 (2022) 083C01.
  * [39] G. Baur, K. Hencken, A. Aste, D. Trautmann, and S. R. Klein, “Multiphoton exchange processes in ultraperipheral relativistic heavy ion collisions”, Nucl. Phys. A 729 (2003) 787–808, arXiv:nucl-th/0307031.
  * [40] F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, “The Color Glass Condensate”, Ann. Rev. Nucl. Part. Sci. 60 (2010) 463–489, arXiv:1002.0333 [hep-ph].
  * [41] C. A. Bertulani and G. Baur, “Electromagnetic Processes in Relativistic Heavy Ion Collisions”, Phys. Rep. 163 (1988) 299.
  * [42] A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, “Heavy ion partial beam lifetimes due to Coulomb induced processes”, Phys. Rev. E 54 (1996) 4233–4239.
  * [43] A. J. Baltz and M. Strikman, “Suppression of heavy ion $\gamma\gamma$ production of the Higgs particle by Coulomb dissociation”, Phys. Rev. D 57 (1998) 548–549, arXiv:hep-ph/9705220.
  * [44] H. Mäntysaari, F. Salazar, B. Schenke, C. Shen, and W. Zhao, “Effects of nuclear structure and quantum interference on diffractive vector meson production in ultraperipheral nuclear collisions”, Phys. Rev. C 109 (2024) 024908, arXiv:2310.15300 [nucl-th].
  * [45] J. Nystrand, A. J. Baltz, and S. R. Klein, “Aspects of Coulomb dissociation and interference in peripheral nucleus-nucleus collisions”, in Workshop on Electromagnetic Probes of Fundamental Physics, pp. 161–170.  3, 2002.  arXiv:nucl-th/0203062.

##  Appendix A The ALICE Collaboration

S. Acharya 127, D. Adamová 86, A. Agarwal135, G. Aglieri Rinella 32, L.
Aglietta24, M. Agnello 29, N. Agrawal 25, Z. Ahammed 135, S. Ahmad 15, S.U.
Ahn 71, I. Ahuja 37, A. Akindinov 141, V. Akishina38, M. Al-Turany 97, D.
Aleksandrov 141, B. Alessandro 56, H.M. Alfanda 6, R. Alfaro Molina 67, B. Ali
15, A. Alici 25, N. Alizadehvandchali 116, A. Alkin 104, J. Alme 20, G. Alocco
52, T. Alt 64, A.R. Altamura 50, I. Altsybeev 95, J.R. Alvarado 44, M.N. Anaam
6, C. Andrei 45, N. Andreou 115, A. Andronic 126, E. Andronov 141, V. Anguelov
94, F. Antinori 54, P. Antonioli 51, N. Apadula 74, L. Aphecetche 103, H.
Appelshäuser 64, C. Arata 73, S. Arcelli 25, M. Aresti 22, R. Arnaldi 56,
J.G.M.C.A. Arneiro 110, I.C. Arsene 19, M. Arslandok 138, A. Augustinus 32, R.
Averbeck 97, M.D. Azmi 15, H. Baba124, A. Badalà 53, J. Bae 104, Y.W. Baek 40,
X. Bai 120, R. Bailhache 64, Y. Bailung 48, R. Bala 91, A. Balbino 29, A.
Baldisseri 130, B. Balis 2, D. Banerjee 4, Z. Banoo 91, V. Barbasova37, F.
Barile 31, L. Barioglio 56, M. Barlou78, B. Barman41, G.G. Barnaföldi 46, L.S.
Barnby 115, E. Barreau 103, V. Barret 127, L. Barreto 110, C. Bartels 119, K.
Barth 32, E. Bartsch 64, N. Bastid 127, S. Basu 75, G. Batigne 103, D.
Battistini 95, B. Batyunya 142, D. Bauri47, J.L. Bazo Alba 101, I.G. Bearden
83, C. Beattie 138, P. Becht 97, D. Behera 48, I. Belikov 129, A.D.C. Bell
Hechavarria 126, F. Bellini 25, R. Bellwied 116, S. Belokurova 141, L.G.E.
Beltran 109, Y.A.V. Beltran 44, G. Bencedi 46, A. Bensaoula116, S. Beole 24,
Y. Berdnikov 141, A. Berdnikova 94, L. Bergmann 94, M.G. Besoiu 63, L. Betev
32, P.P. Bhaduri 135, A. Bhasin 91, B. Bhattacharjee 41, L. Bianchi 24, N.
Bianchi 49, J. Bielčík 35, J. Bielčíková 86, A.P. Bigot 129, A. Bilandzic 95,
G. Biro 46, S. Biswas 4, N. Bize 103, J.T. Blair 108, D. Blau 141, M.B.
Blidaru 97, N. Bluhme38, C. Blume 64, G. Boca 21,55, F. Bock 87, T. Bodova 20,
J. Bok 16, L. Boldizsár 46, M. Bombara 37, P.M. Bond 32, G. Bonomi 134,55, H.
Borel 130, A. Borissov 141, A.G. Borquez Carcamo 94, H. Bossi 138, E. Botta
24, Y.E.M. Bouziani 64, L. Bratrud 64, P. Braun-Munzinger 97, M. Bregant 110,
M. Broz 35, G.E. Bruno 96,31, V.D. Buchakchiev 36, M.D. Buckland 23, D.
Budnikov 141, H. Buesching 64, S. Bufalino 29, P. Buhler 102, N. Burmasov 141,
Z. Buthelezi 68,123, A. Bylinkin 20, S.A. Bysiak107, J.C. Cabanillas Noris
109, M.F.T. Cabrera116, M. Cai 6, H. Caines 138, A. Caliva 28, E. Calvo Villar
101, J.M.M. Camacho 109, P. Camerini 23, F.D.M. Canedo 110, S.L. Cantway 138,
M. Carabas 113, A.A. Carballo 32, F. Carnesecchi 32, R. Caron 128, L.A.D.
Carvalho 110, J. Castillo Castellanos 130, M. Castoldi 32, F. Catalano 32, S.
Cattaruzzi 23, C. Ceballos Sanchez 142, R. Cerri24, I. Chakaberia 74, P.
Chakraborty 136,47, S. Chandra 135, S. Chapeland 32, M. Chartier 119, S.
Chattopadhay135, S. Chattopadhyay 135, S. Chattopadhyay 99, M. Chen39, T.
Cheng 97,6, C. Cheshkov 128, V. Chibante Barroso 32, D.D. Chinellato 111, E.S.
Chizzali II,95, J. Cho 58, S. Cho 58, P. Chochula 32, Z.A. Chochulska136, D.
Choudhury41, P. Christakoglou 84, C.H. Christensen 83, P. Christiansen 75, T.
Chujo 125, M. Ciacco 29, C. Cicalo 52, M.R. Ciupek97, G. ClaiIII,51, F.
Colamaria 50, J.S. Colburn100, D. Colella 31, M. Colocci 25, M. Concas 32, G.
Conesa Balbastre 73, Z. Conesa del Valle 131, G. Contin 23, J.G. Contreras 35,
M.L. Coquet 103,130, P. Cortese 133,56, M.R. Cosentino 112, F. Costa 32, S.
Costanza 21,55, C. Cot 131, J. Crkovská 94, P. Crochet 127, R. Cruz-Torres 74,
P. Cui 6, M.M. Czarnynoga136, A. Dainese 54, G. Dange38, M.C. Danisch 94, A.
Danu 63, P. Das 80, P. Das 4, S. Das 4, A.R. Dash 126, S. Dash 47, A. De Caro
28, G. de Cataldo 50, J. de Cuveland38, A. De Falco 22, D. De Gruttola 28, N.
De Marco 56, C. De Martin 23, S. De Pasquale 28, R. Deb 134, R. Del Grande 95,
L. Dello Stritto 32, W. Deng 6, K.C. Devereaux18, P. Dhankher 18, D. Di Bari
31, A. Di Mauro 32, B. Diab 130, R.A. Diaz 142,7, T. Dietel 114, Y. Ding 6, J.
Ditzel 64, R. Divià 32, Ø. Djuvsland20, U. Dmitrieva 141, A. Dobrin 63, B.
Dönigus 64, J.M. Dubinski 136, A. Dubla 97, P. Dupieux 127, N. Dzalaiova13,
T.M. Eder 126, R.J. Ehlers 74, F. Eisenhut 64, R. Ejima92, D. Elia 50, B.
Erazmus 103, F. Ercolessi 25, B. Espagnon 131, G. Eulisse 32, D. Evans 100, S.
Evdokimov 141, L. Fabbietti 95, M. Faggin 23, J. Faivre 73, F. Fan 6, W. Fan
74, A. Fantoni 49, M. Fasel 87, A. Feliciello 56, G. Feofilov 141, A.
Fernández Téllez 44, L. Ferrandi 110, M.B. Ferrer 32, A. Ferrero 130, C.
Ferrero IV,56, A. Ferretti 24, V.J.G. Feuillard 94, V. Filova 35, D. Finogeev
141, F.M. Fionda 52, E. Flatland32, F. Flor 138,116, A.N. Flores 108, S.
Foertsch 68, I. Fokin 94, S. Fokin 141, U. FolloIV,56, E. Fragiacomo 57, E.
Frajna 46, U. Fuchs 32, N. Funicello 28, C. Furget 73, A. Furs 141, T.
Fusayasu 98, J.J. Gaardhøje 83, M. Gagliardi 24, A.M. Gago 101, T. Gahlaut47,
C.D. Galvan 109, D.R. Gangadharan 116, P. Ganoti 78, C. Garabatos 97, J.M.
Garcia44, T. García Chávez 44, E. Garcia-Solis 9, C. Gargiulo 32, P. Gasik 97,
H.M. Gaur38, A. Gautam 118, M.B. Gay Ducati 66, M. Germain 103, C. Ghosh135,
M. Giacalone 51, G. Gioachin 29, P. Giubellino 97,56, P. Giubilato 27, A.M.C.
Glaenzer 130, P. Glässel 94, E. Glimos 122, D.J.Q. Goh76, V. Gonzalez 137, P.
Gordeev 141, M. Gorgon 2, K. Goswami 48, S. Gotovac33, V. Grabski 67, L.K.
Graczykowski 136, E. Grecka 86, A. Grelli 59, C. Grigoras 32, V. Grigoriev
141, S. Grigoryan 142,1, F. Grosa 32, J.F. Grosse-Oetringhaus 32, R. Grosso
97, D. Grund 35, N.A. Grunwald94, G.G. Guardiano 111, R. Guernane 73, M.
Guilbaud 103, K. Gulbrandsen 83, J.J.W.K. Gumprecht102, T. Gündem 64, T. Gunji
124, W. Guo 6, A. Gupta 91, R. Gupta 91, R. Gupta 48, K. Gwizdziel 136, L.
Gyulai 46, C. Hadjidakis 131, F.U. Haider 91, S. Haidlova 35, M. Haldar4, H.
Hamagaki 76, A. Hamdi 74, Y. Han 139, B.G. Hanley 137, R. Hannigan 108, J.
Hansen 75, M.R. Haque 97, J.W. Harris 138, A. Harton 9, M.V. Hartung 64, H.
Hassan 117, D. Hatzifotiadou 51, P. Hauer 42, L.B. Havener 138, E. Hellbär 97,
H. Helstrup 34, M. Hemmer 64, T. Herman 35, S.G. Hernandez116, G. Herrera
Corral 8, S. Herrmann 128, K.F. Hetland 34, B. Heybeck 64, H. Hillemanns 32,
B. Hippolyte 129, F.W. Hoffmann 70, B. Hofman 59, G.H. Hong 139, M. Horst 95,
A. Horzyk 2, Y. Hou 6, P. Hristov 32, P. Huhn64, L.M. Huhta 117, T.J. Humanic
88, A. Hutson 116, D. Hutter 38, M.C. Hwang 18, R. Ilkaev141, M. Inaba 125,
G.M. Innocenti 32, M. Ippolitov 141, A. Isakov 84, T. Isidori 118, M.S. Islam
99, S. Iurchenko141, M. Ivanov 97, M. Ivanov13, V. Ivanov 141, K.E. Iversen
75, M. Jablonski 2, B. Jacak 18,74, N. Jacazio 25, P.M. Jacobs 74, S.
Jadlovska106, J. Jadlovsky106, S. Jaelani 82, C. Jahnke 110, M.J. Jakubowska
136, M.A. Janik 136, T. Janson70, S. Ji 16, S. Jia 10, F. Jonas 74, D.M. Jones
119, J.M. Jowett 32,97, J. Jung 64, M. Jung 64, A. Junique 32, A. Jusko 100,
J. Kaewjai105, P. Kalinak 60, A. Kalweit 32, A. Karasu Uysal V,72, D.
Karatovic 89, N. Karatzenis100, O. Karavichev 141, T. Karavicheva 141, E.
Karpechev 141, M.J. Karwowska 32,136, U. Kebschull 70, R. Keidel 140, M. Keil
32, B. Ketzer 42, S.S. Khade 48, A.M. Khan 120, S. Khan 15, A. Khanzadeev 141,
Y. Kharlov 141, A. Khatun 118, A. Khuntia 35, Z. Khuranova 64, B. Kileng 34,
B. Kim 104, C. Kim 16, D.J. Kim 117, E.J. Kim 69, J. Kim 139, J. Kim 58, J.
Kim 32,69, M. Kim 18, S. Kim 17, T. Kim 139, K. Kimura 92, A. Kirkova36, S.
Kirsch 64, I. Kisel 38, S. Kiselev 141, A. Kisiel 136, J.P. Kitowski 2, J.L.
Klay 5, J. Klein 32, S. Klein 74, C. Klein-Bösing 126, M. Kleiner 64, T.
Klemenz 95, A. Kluge 32, C. Kobdaj 105, R. Kohara124, T. Kollegger97, A.
Kondratyev 142, N. Kondratyeva 141, J. Konig 64, S.A. Konigstorfer 95, P.J.
Konopka 32, G. Kornakov 136, M. Korwieser 95, S.D. Koryciak 2, C. Koster84, A.
Kotliarov 86, N. Kovacic89, V. Kovalenko 141, M. Kowalski 107, V. Kozhuharov
36, I. Králik 60, A. Kravčáková 37, L. Krcal 32,38, M. Krivda 100,60, F.
Krizek 86, K. Krizkova Gajdosova 32, C. Krug 66, M. Krüger 64, D.M. Krupova
35, E. Kryshen 141, V. Kučera 58, C. Kuhn 129, P.G. Kuijer 84, T. Kumaoka125,
D. Kumar135, L. Kumar 90, N. Kumar90, S. Kumar 31, S. Kundu 32, P. Kurashvili
79, A. Kurepin 141, A.B. Kurepin 141, A. Kuryakin 141, S. Kushpil 86, V.
Kuskov 141, M. Kutyla136, A. Kuznetsov142, M.J. Kweon 58, Y. Kwon 139, S.L. La
Pointe 38, P. La Rocca 26, A. Lakrathok105, M. Lamanna 32, A.R. Landou 73, R.
Langoy 121, P. Larionov 32, E. Laudi 32, L. Lautner 32,95, R.A.N. Laveaga109,
R. Lavicka 102, R. Lea 134,55, H. Lee 104, I. Legrand 45, G. Legras 126, J.
Lehrbach 38, A.M. Lejeune35, T.M. Lelek2, R.C. Lemmon 85, I. León Monzón 109,
M.M. Lesch 95, E.D. Lesser 18, P. Lévai 46, M. Li6, X. Li10, B.E. Liang-gilman
18, J. Lien 121, R. Lietava 100, I. Likmeta 116, B. Lim 24, S.H. Lim 16, V.
Lindenstruth 38, A. Lindner45, C. Lippmann 97, D.H. Liu 6, J. Liu 119, G.S.S.
Liveraro 111, I.M. Lofnes 20, C. Loizides 87, S. Lokos 107, J. Lömker 59, X.
Lopez 127, E. López Torres 7, P. Lu 97,120, F.V. Lugo 67, J.R. Luhder 126, M.
Lunardon 27, G. Luparello 57, Y.G. Ma 39, M. Mager 32, A. Maire 129, E.M.
Majerz2, M.V. Makariev 36, M. Malaev 141, G. Malfattore 25, N.M. Malik 91,
Q.W. Malik19, S.K. Malik 91, L. Malinina I,VIII,142, D. Mallick 131, N.
Mallick 48, G. Mandaglio 30,53, S.K. Mandal 79, A. Manea 63, V. Manko 141, F.
Manso 127, V. Manzari 50, Y. Mao 6, R.W. Marcjan 2, G.V. Margagliotti 23, A.
Margotti 51, A. Marín 97, C. Markert 108, P. Martinengo 32, M.I. Martínez 44,
G. Martínez García 103, M.P.P. Martins 110, S. Masciocchi 97, M. Masera 24, A.
Masoni 52, L. Massacrier 131, O. Massen 59, A. Mastroserio 132,50, O. Matonoha
75, S. Mattiazzo 27, A. Matyja 107, A.L. Mazuecos 32, F. Mazzaschi 32,24, M.
Mazzilli 116, J.E. Mdhluli 123, Y. Melikyan 43, A. Menchaca-Rocha 67, J.E.M.
Mendez 65, E. Meninno 102, A.S. Menon 116, M.W. Menzel32,94, M. Meres 13, Y.
Miake125, L. Micheletti 32, D.L. Mihaylov 95, K. Mikhaylov 142,141, N. Minafra
118, D. Miśkowiec 97, A. Modak 134,4, B. Mohanty80, M. Mohisin Khan VI,15,
M.A. Molander 43, S. Monira 136, C. Mordasini 117, D.A. Moreira De Godoy 126,
I. Morozov 141, A. Morsch 32, T. Mrnjavac 32, V. Muccifora 49, S. Muhuri 135,
J.D. Mulligan 74, A. Mulliri 22, M.G. Munhoz 110, R.H. Munzer 64, H. Murakami
124, S. Murray 114, L. Musa 32, J. Musinsky 60, J.W. Myrcha 136, B. Naik 123,
A.I. Nambrath 18, B.K. Nandi 47, R. Nania 51, E. Nappi 50, A.F. Nassirpour 17,
A. Nath 94, C. Nattrass 122, M.N. Naydenov 36, A. Neagu19, A. Negru113, E.
Nekrasova141, L. Nellen 65, R. Nepeivoda 75, S. Nese 19, G. Neskovic 38, N.
Nicassio 50, B.S. Nielsen 83, E.G. Nielsen 83, S. Nikolaev 141, S. Nikulin
141, V. Nikulin 141, F. Noferini 51, S. Noh 12, P. Nomokonov 142, J. Norman
119, N. Novitzky 87, P. Nowakowski 136, A. Nyanin 141, J. Nystrand 20, S. Oh
17, A. Ohlson 75, V.A. Okorokov 141, J. Oleniacz 136, A. Onnerstad 117, C.
Oppedisano 56, A. Ortiz Velasquez 65, J. Otwinowski 107, M. Oya92, K. Oyama
76, Y. Pachmayer 94, S. Padhan 47, D. Pagano 134,55, G. Paić 65, S. Paisano-
Guzmán 44, A. Palasciano 50, S. Panebianco 130, H. Park 125, H. Park 104, J.
Park 125, J.E. Parkkila 32, Y. Patley 47, B. Paul 22, M.M.D.M. Paulino 110, H.
Pei 6, T. Peitzmann 59, X. Peng 11, M. Pennisi 24, S. Perciballi 24, D.
Peresunko 141, G.M. Perez 7, Y. Pestov141, M.T. Petersen83, V. Petrov 141, M.
Petrovici 45, S. Piano 57, M. Pikna 13, P. Pillot 103, O. Pinazza 51,32, L.
Pinsky116, C. Pinto 95, S. Pisano 49, M. Płoskoń 74, M. Planinic89, F.
Pliquett64, M.G. Poghosyan 87, B. Polichtchouk 141, S. Politano 29, N. Poljak
89, A. Pop 45, S. Porteboeuf-Houssais 127, V. Pozdniakov I,142, I.Y. Pozos 44,
K.K. Pradhan 48, S.K. Prasad 4, S. Prasad 48, R. Preghenella 51, F. Prino 56,
C.A. Pruneau 137, I. Pshenichnov 141, M. Puccio 32, S. Pucillo 24, S. Qiu 84,
L. Quaglia 24, S. Ragoni 14, A. Rai 138, A. Rakotozafindrabe 130, L. Ramello
133,56, F. Rami 129, M. Rasa 26, S.S. Räsänen 43, R. Rath 51, M.P. Rauch 20,
I. Ravasenga 32, K.F. Read 87,122, C. Reckziegel 112, A.R. Redelbach 38, K.
Redlich VII,79, C.A. Reetz 97, H.D. Regules-Medel44, A. Rehman20, F. Reidt 32,
H.A. Reme-Ness 34, Z. Rescakova37, K. Reygers 94, A. Riabov 141, V. Riabov
141, R. Ricci 28, M. Richter 20, A.A. Riedel 95, W. Riegler 32, A.G. Riffero
24, C. Ripoli28, C. Ristea 63, M.V. Rodriguez 32, M. Rodríguez Cahuantzi 44,
S.A. Rodríguez Ramírez 44, K. Røed 19, R. Rogalev 141, E. Rogochaya 142, T.S.
Rogoschinski 64, D. Rohr 32, D. Röhrich 20, S. Rojas Torres 35, P.S. Rokita
136, G. Romanenko 25, F. Ronchetti 49, E.D. Rosas65, K. Roslon 136, A. Rossi
54, A. Roy 48, S. Roy 47, N. Rubini 25, J.A. Rudolph84, D. Ruggiano 136, R.
Rui 23, P.G. Russek 2, R. Russo 84, A. Rustamov 81, E. Ryabinkin 141, Y.
Ryabov 141, A. Rybicki 107, J. Ryu 16, W. Rzesa 136, S. Sadhu 31, S. Sadovsky
141, J. Saetre 20, K. Šafařík 35, S.K. Saha 4, S. Saha 80, B. Sahoo 48, R.
Sahoo 48, S. Sahoo61, D. Sahu 48, P.K. Sahu 61, J. Saini 135, K. Sajdakova37,
S. Sakai 125, M.P. Salvan 97, S. Sambyal 91, D. Samitz 102, I. Sanna 32,95,
T.B. Saramela110, D. Sarkar 83, P. Sarma 41, V. Sarritzu 22, V.M. Sarti 95,
M.H.P. Sas 32, S. Sawan 80, E. Scapparone 51, J. Schambach 87, H.S. Scheid 64,
C. Schiaua 45, R. Schicker 94, F. Schlepper 94, A. Schmah97, C. Schmidt 97,
H.R. Schmidt93, M.O. Schmidt 32, M. Schmidt93, N.V. Schmidt 87, A.R. Schmier
122, R. Schotter 129, A. Schröter 38, J. Schukraft 32, K. Schweda 97, G.
Scioli 25, E. Scomparin 56, J.E. Seger 14, Y. Sekiguchi124, D. Sekihata 124,
M. Selina 84, I. Selyuzhenkov 97, S. Senyukov 129, J.J. Seo 94, D. Serebryakov
141, L. Serkin 65, L. Šerkšnytė 95, A. Sevcenco 63, T.J. Shaba 68, A. Shabetai
103, R. Shahoyan32, A. Shangaraev 141, B. Sharma 91, D. Sharma 47, H. Sharma
54, M. Sharma 91, S. Sharma 76, S. Sharma 91, U. Sharma 91, A. Shatat 131, O.
Sheibani116, K. Shigaki 92, M. Shimomura77, J. Shin12, S. Shirinkin 141, Q.
Shou 39, Y. Sibiriak 141, S. Siddhanta 52, T. Siemiarczuk 79, T.F. Silva 110,
D. Silvermyr 75, T. Simantathammakul105, R. Simeonov 36, B. Singh91, B. Singh
95, K. Singh 48, R. Singh 80, R. Singh 91, R. Singh 97,48, S. Singh 15, V.K.
Singh 135, V. Singhal 135, T. Sinha 99, B. Sitar 13, M. Sitta 133,56, T.B.
Skaali19, G. Skorodumovs 94, N. Smirnov 138, R.J.M. Snellings 59, E.H. Solheim
19, J. Song 16, C. Sonnabend 32,97, J.M. Sonneveld 84, F. Soramel 27, A.B.
Soto-hernandez 88, R. Spijkers 84, I. Sputowska 107, J. Staa 75, J. Stachel
94, I. Stan 63, P.J. Steffanic 122, S.F. Stiefelmaier 94, D. Stocco 103, I.
Storehaug 19, N.J. Strangmann 64, P. Stratmann 126, S. Strazzi 25, A.
Sturniolo 30,53, C.P. Stylianidis84, A.A.P. Suaide 110, C. Suire 131, M.
Sukhanov 141, M. Suljic 32, R. Sultanov 141, V. Sumberia 91, S. Sumowidagdo
82, I. Szarka 13, M. Szymkowski 136, S.F. Taghavi 95, G. Taillepied 97, J.
Takahashi 111, G.J. Tambave 80, S. Tang 6, Z. Tang 120, J.D. Tapia Takaki 118,
N. Tapus113, L.A. Tarasovicova 126, M.G. Tarzila 45, G.F. Tassielli 31, A.
Tauro 32, A. Tavira García 131, G. Tejeda Muñoz 44, A. Telesca 32, L. Terlizzi
24, C. Terrevoli 50, S. Thakur 4, D. Thomas 108, A. Tikhonov 141, N. Tiltmann
32,126, A.R. Timmins 116, M. Tkacik106, T. Tkacik 106, A. Toia 64, R.
Tokumoto92, S. Tomassini25, K. Tomohiro92, N. Topilskaya 141, M. Toppi 49, T.
Tork 131, V.V. Torres 103, A.G. Torres Ramos 31, A. Trifiró 30,53, T.
Triloki96, A.S. Triolo 32,30,53, S. Tripathy 32, T. Tripathy 47, V. Trubnikov
3, W.H. Trzaska 117, T.P. Trzcinski 136, C. Tsolanta19, R. Tu39, A. Tumkin
141, R. Turrisi 54, T.S. Tveter 19, K. Ullaland 20, B. Ulukutlu 95, A. Uras
128, M. Urioni 134, G.L. Usai 22, M. Vala37, N. Valle 55, L.V.R. van
Doremalen59, M. van Leeuwen 84, C.A. van Veen 94, R.J.G. van Weelden 84, P.
Vande Vyvre 32, D. Varga 46, Z. Varga 46, P. Vargas Torres65, M. Vasileiou 78,
A. Vasiliev 141, O. Vázquez Doce 49, O. Vazquez Rueda 116, V. Vechernin 141,
E. Vercellin 24, S. Vergara Limón44, R. Verma47, L. Vermunt 97, R. Vértesi 46,
M. Verweij 59, L. Vickovic33, Z. Vilakazi123, O. Villalobos Baillie 100, A.
Villani 23, A. Vinogradov 141, T. Virgili 28, M.M.O. Virta 117, V.
Vislavicius75, A. Vodopyanov 142, B. Volkel 32, M.A. Völkl 94, S.A. Voloshin
137, G. Volpe 31, B. von Haller 32, I. Vorobyev 32, N. Vozniuk 141, J.
Vrláková 37, J. Wan39, C. Wang 39, D. Wang39, Y. Wang 39, Y. Wang 6, A.
Wegrzynek 32, F.T. Weiglhofer38, S.C. Wenzel 32, J.P. Wessels 126, J. Wiechula
64, J. Wikne 19, G. Wilk 79, J. Wilkinson 97, G.A. Willems 126, B. Windelband
94, M. Winn 130, J.R. Wright 108, W. Wu39, Y. Wu 120, Z. Xiong120, R. Xu 6, A.
Yadav 42, A.K. Yadav 135, Y. Yamaguchi 92, S. Yang20, S. Yano 92, E.R.
Yeats18, Z. Yin 6, I.-K. Yoo 16, J.H. Yoon 58, H. Yu12, S. Yuan20, A. Yuncu
94, V. Zaccolo 23, C. Zampolli 32, M. Zang6, F. Zanone 94, N. Zardoshti 32, A.
Zarochentsev 141, P. Závada 62, N. Zaviyalov141, M. Zhalov 141, B. Zhang 6, C.
Zhang 130, L. Zhang 39, M. Zhang127,6, S. Zhang 39, X. Zhang 6, Y. Zhang120,
Z. Zhang 6, M. Zhao 10, V. Zherebchevskii 141, Y. Zhi10, D. Zhou 6, Y. Zhou
83, J. Zhu 54,6, S. Zhu120, Y. Zhu6, S.C. Zugravel 56, N. Zurlo 134,55

## Affiliation Notes

I Deceased
II Also at: Max-Planck-Institut fur Physik, Munich, Germany
III Also at: Italian National Agency for New Technologies, Energy and
Sustainable Economic Development (ENEA), Bologna, Italy
IV Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy
V Also at: Yildiz Technical University, Istanbul, Türkiye
VI Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh,
India
VII Also at: Institute of Theoretical Physics, University of Wroclaw, Poland
VIII Also at: An institution covered by a cooperation agreement with CERN

## Collaboration Institutes

1 A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
2 AGH University of Krakow, Cracow, Poland
3 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine
4 Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
5 California Polytechnic State University, San Luis Obispo, California, United
States
6 Central China Normal University, Wuhan, China
7 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
8 Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico
9 Chicago State University, Chicago, Illinois, United States
10 China Institute of Atomic Energy, Beijing, China
11 China University of Geosciences, Wuhan, China
12 Chungbuk National University, Cheongju, Republic of Korea
13 Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovak Republic
14 Creighton University, Omaha, Nebraska, United States
15 Department of Physics, Aligarh Muslim University, Aligarh, India
16 Department of Physics, Pusan National University, Pusan, Republic of Korea
17 Department of Physics, Sejong University, Seoul, Republic of Korea
18 Department of Physics, University of California, Berkeley, California,
United States
19 Department of Physics, University of Oslo, Oslo, Norway
20 Department of Physics and Technology, University of Bergen, Bergen, Norway
21 Dipartimento di Fisica, Università di Pavia, Pavia, Italy
22 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
23 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
24 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
25 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
28 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
29 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
30 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy
31 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
32 European Organization for Nuclear Research (CERN), Geneva, Switzerland
33 Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia
34 Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway
35 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
36 Faculty of Physics, Sofia University, Sofia, Bulgaria
37 Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic
38 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
39 Fudan University, Shanghai, China
40 Gangneung-Wonju National University, Gangneung, Republic of Korea
41 Gauhati University, Department of Physics, Guwahati, India
42 Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
43 Helsinki Institute of Physics (HIP), Helsinki, Finland
44 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico
45 Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania
46 HUN-REN Wigner Research Centre for Physics, Budapest, Hungary
47 Indian Institute of Technology Bombay (IIT), Mumbai, India
48 Indian Institute of Technology Indore, Indore, India
49 INFN, Laboratori Nazionali di Frascati, Frascati, Italy
50 INFN, Sezione di Bari, Bari, Italy
51 INFN, Sezione di Bologna, Bologna, Italy
52 INFN, Sezione di Cagliari, Cagliari, Italy
53 INFN, Sezione di Catania, Catania, Italy
54 INFN, Sezione di Padova, Padova, Italy
55 INFN, Sezione di Pavia, Pavia, Italy
56 INFN, Sezione di Torino, Turin, Italy
57 INFN, Sezione di Trieste, Trieste, Italy
58 Inha University, Incheon, Republic of Korea
59 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University/Nikhef, Utrecht, Netherlands
60 Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovak Republic
61 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India
62 Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
63 Institute of Space Science (ISS), Bucharest, Romania
64 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
65 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
66 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil
67 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico
68 iThemba LABS, National Research Foundation, Somerset West, South Africa
69 Jeonbuk National University, Jeonju, Republic of Korea
70 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany
71 Korea Institute of Science and Technology Information, Daejeon, Republic of
Korea
72 KTO Karatay University, Konya, Turkey
73 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France
74 Lawrence Berkeley National Laboratory, Berkeley, California, United States
75 Lund University Department of Physics, Division of Particle Physics, Lund,
Sweden
76 Nagasaki Institute of Applied Science, Nagasaki, Japan
77 Nara Women’s University (NWU), Nara, Japan
78 National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece
79 National Centre for Nuclear Research, Warsaw, Poland
80 National Institute of Science Education and Research, Homi Bhabha National
Institute, Jatni, India
81 National Nuclear Research Center, Baku, Azerbaijan
82 National Research and Innovation Agency - BRIN, Jakarta, Indonesia
83 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
84 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands
85 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
86 Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež,
Czech Republic
87 Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
88 Ohio State University, Columbus, Ohio, United States
89 Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia
90 Physics Department, Panjab University, Chandigarh, India
91 Physics Department, University of Jammu, Jammu, India
92 Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan
93 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany
94 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
95 Physik Department, Technische Universität München, Munich, Germany
96 Politecnico di Bari and Sezione INFN, Bari, Italy
97 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Darmstadt, Germany
98 Saga University, Saga, Japan
99 Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata,
India
100 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
101 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru
102 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
103 SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France
104 Sungkyunkwan University, Suwon City, Republic of Korea
105 Suranaree University of Technology, Nakhon Ratchasima, Thailand
106 Technical University of Košice, Košice, Slovak Republic
107 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland
108 The University of Texas at Austin, Austin, Texas, United States
109 Universidad Autónoma de Sinaloa, Culiacán, Mexico
110 Universidade de São Paulo (USP), São Paulo, Brazil
111 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
112 Universidade Federal do ABC, Santo Andre, Brazil
113 Universitatea Nationala de Stiinta si Tehnologie Politehnica Bucuresti,
Bucharest, Romania
114 University of Cape Town, Cape Town, South Africa
115 University of Derby, Derby, United Kingdom
116 University of Houston, Houston, Texas, United States
117 University of Jyväskylä, Jyväskylä, Finland
118 University of Kansas, Lawrence, Kansas, United States
119 University of Liverpool, Liverpool, United Kingdom
120 University of Science and Technology of China, Hefei, China
121 University of South-Eastern Norway, Kongsberg, Norway
122 University of Tennessee, Knoxville, Tennessee, United States
123 University of the Witwatersrand, Johannesburg, South Africa
124 University of Tokyo, Tokyo, Japan
125 University of Tsukuba, Tsukuba, Japan
126 Universität Münster, Institut für Kernphysik, Münster, Germany
127 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
128 Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de
Lyon, Lyon, France
129 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France,
Strasbourg, France
130 Université Paris-Saclay, Centre d’Etudes de Saclay (CEA), IRFU, Départment
de Physique Nucléaire (DPhN), Saclay, France
131 Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
132 Università degli Studi di Foggia, Foggia, Italy
133 Università del Piemonte Orientale, Vercelli, Italy
134 Università di Brescia, Brescia, Italy
135 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata,
India
136 Warsaw University of Technology, Warsaw, Poland
137 Wayne State University, Detroit, Michigan, United States
138 Yale University, New Haven, Connecticut, United States
139 Yonsei University, Seoul, Republic of Korea
140 Zentrum für Technologie und Transfer (ZTT), Worms, Germany
141 Affiliated with an institute covered by a cooperation agreement with CERN
142 Affiliated with an international laboratory covered by a cooperation
agreement with CERN.


# Electronic and thermoelectric properties of
quasi-fractal carbon nitride nanoribbons

R.T. Sibatov1,2, Alireza Khalili Golmankhaneh3,
R.M. Meftakhutdinov4, E.V. Morozova4, D.A. Timkaeva4

###### Abstract

Recent works devoted to the synthesis of artificial molecular systems with
quasi-fractal geometry provide new opportunities for the experimental study of
electronic properties in atomic systems of fractional dimension. There has
been a renewed interest in theoretical studies of transport phenomena in
fractal nanosystems. In this work, using calculations based on the density
functional theory, molecular dynamics, and the method of non-equilibrium Green
functions, we study electronic and thermoelectric properties of a device based
on quasi-fractal carbon nitride nanoribbon with Sierpinski triangle blocks. We
estimate changes in transport properties with generation $g$ of quasi-fractal
blocks in nanoribbon.

1 Moscow Institute of Physics and Technology (MIPT), Dolgoprudny 141700,
Russia

2 Scientific-Manufacturing Complex ‘‘Technological Centre’’, Moscow 124498,
Russia

3 Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran

4 Ulyanovsk State University, Ulyanovsk 432017, Russia

##  1 Introduction

The dimension of the electronic quantum system largely determines its
properties. Particularly, in one-dimensional systems electrons form the
Luttinger liquid [1], and in two-dimensional systems the quantum Hall effect
is observed [2]. Little is known about the behavior of electrons in fractional
dimensional systems [3]. Recent works [4, 3, 5] devoted to the synthesis of
artificial molecular systems with quasi-fractal geometry provide new
opportunities for experimental study of electronic properties in atomic
systems of fractional dimension. The molecular quasi-fractal in the form of
Sierpinski triangle was experimentally obtained in [5] by self-assembly of
organic molecules by halogen bonds, hydrogen bonds, covalent bonds, and
coordination interactions of metal-organic compounds on the surface. The
resulting structure contains pores of different sizes in one triangular block
and, according to forecasts, it has unique optical, magnetic and mechanical
properties.

In works [6, 7, 8], the first principles calculations and semi-empirical
methods are used for the theoretical study of electronic properties of
fractional-sized molecular systems. In [8], electronic, optical, and
magnetooptic properties of the Sierpinski fractal triangles built on a
graphene sheet are studied within the Hubbard model. Significant differences
in the properties of structures with ‘‘zigzag’’ and ‘‘armchair’’ edges were
found, similar to nanoribbons, quantum dots, and anti-dots. Sierpinski
triangles with zigzag edges have a large proportion of boundary states, that
leads to instability with respect to spin polarization with account for the
electron-electron interaction. On the contrary, fractals with armchair edges
remain balanced on spin. In both cases, there is a strong energy gap, which
leads to a pronounced optical absorption in the visible energy range. It has
been shown that the distribution of energy levels is self-similar for quasi-
fractal sets of later generation. In [9], the authors suggest a fractal
Kronig-Penny model describing the quantum behavior of a particle in fractal
lattice. In [10], the properties of photon transport in fractal grids in the
form of Sierpinski carpet and triangle are experimentally examined, the mean
squared displacement and the Polia number are determined. The authors observe
anomalous diffusion of photons, and determine the dependence of the critical
point of transition from normal to anomalous transfer on the fractal geometry
parameters.

The authors of Ref. [4] implemented the packing of molecular Sierpinski
triangles in one-dimensional crystals by the templates method in ultra-high
vacuum. The obtained structures were studied by means of low-temperature
scanning tunneling microscopy. The states described by electron wave functions
of fractional dimension were observed. The wave functions delocalized over the
Sierpinski structure are spread out on self-similar parts at higher energies,
and this large-scale invariance can also be recovered in the reciprocal space.

Figure 1: Triazine-based structure C3N4 (left panel) and poly-heptazine
structure (tri-s-triazine) (right panel) [11].

We consider nanoribbons constructed from quasi-fractal triangles. The geometry
of these ribbons is similar to the geometry of the experimental structures
demonstrated in [4], but we have chosen carbon nitride structures as
structural blocks. Recently, nitrogen and carbon compounds with high N:C ratio
and graphite polymer structure have been actively explored as potential next-
generation materials for inclusion in energy conversion and storage devices,
as well as for optoelectronic and catalytic applications [11]. Among them,
popular materials consist of C- and N-containing heterocycles with heptazine
or triazine rings bound via sp2-linked nitrogen atoms N(C)3 or –NH– groups
[11, 12] (Fig. 1). In some works (see, e.g., [13]), the authors consider
structures where the central nitrogen atom in a heptazine triangle is
substituted with a carbon atom.

Figure 2: The geometry of devices based on carbon nitride nanoribbons with the
block triangles of first and second generation.

In this paper, we study electron and phonon transport in a device based on a
quasi-fractal carbon nitride nanoribbons with quasi-fractal Sierpinski
triangles, using calculations based on methods of density functional theory
(DFT), molecular dynamics (MD) and non-equilibrium Green functions (NEGF). The
cases of the first and second generations of such ribbons are shown in Fig. 2.
As electrodes, we used graphene nanoribbons attached to the corners of the
studied quasi-fractal ribbons, as shown in Fig. 2. We calculate the electron
and phonon transmission spectra, and then estimate the thermoelectric
properties of the nanoribbons under consideration for different generations
$g$. In the second part of the work, using the Monte Carlo simulation method,
we investigate the properties of hopping transport in quasi-fractal lattices,
determine the mode and parameters of anomalous diffusion in the absence and
presence of an external electric field for different values of localization
radius and levels of energy disorder.

##  2 Electronic properties

Let us first consider the flat quasi-fractal crystalline lattices by
performing their optimization and calculating their electronic band structures
and absorption spectra. These lattices are periodic with an elementary cell,
representing the atomic quasi-fractal Sierpinski triangle of $g$-th
generation. Optimization is performed for quasi-fractal crystalline structures
using the DFT method implemented in Quantum ATK [14]. The cutoff energy of the
electron wave functions is 500 eV. The reliable convergence criteria for total
energy and power are as follows $10^{-6}$ eV and 0.01 eV/Å. The transverse
cell size is 30 Å. We used the Perdue-Burke-Ernzerhoff approximation to
describe the effects of electronic exchange and correlation, to optimize the
structure and to calculate electronic properties. The Monkhorst–Pack method
was used to generate $k$ points in the Brillouin zone.

Fig. 3 shows the band structures of the periodic nitrogen-carbon lattices, in
which the unit cell contains a quasifractal Sierpinski triangle of $g$-th
generation. Fig. 4 shows the absorption spectra of the corresponding
structures. To the right of Fig. 4, a unit cell for case $g=3$ is shown. In
the crystal with $g=3$, there are triangular pores of three sizes. With the
increase of $g$ for the elementary blocks of the crystal structure, the
flattening of the branches is observed and, consequently, the increase in the
effective masses of the charge carriers takes place. Individual zones for
large $g$ become simple lines of the molecular system due to the localization
of the carriers in fragments of the quasi-fractal blocks.

Figure 3: Band structures of quasi-fractal lattices for $g=2$ (a), $g=3$ (b),
$g=4$ (c).

Usually fractal systems are characterized by a wider absorption frequency
range. In some sense, our results contradict this view. The absorption spectra
in Fig. 4 indicate that with an increase in $g$ not only the absorption index
decreases. The absorption bands are constricted and the corresponding
structures are characterized by a narrower absorption range. This is due to
the fact that, in comparison with macroscopic fractals, in atomistic
structures, along with a change in geometry, the electronic states of the
system change. The spectrum of the quasi-fractal system under consideration
tends to go from the spectrum of the periodic 2d system to the spectrum of an
isolated molecular system.

Figure 4: Absorption spectra of planar quasi-fractal crystals for three
generations. The right panel shows the structure for the case $g=3$; the
crystal contains triangular pores of three sizes. Figure 5: Real and imaginary
parts of the optical conductivity of quasi-fractal gratings for $g=2$ (a),
$g=3$ (b), $g=4$ (c).

Fig. 5 shows the frequency dependencies of the real and imaginary parts of the
optical conductivity for the investigated quasi-fractal crystal structures. As
the generation $g$ increases, the optical conductivity decreases and the
nonzero conduction bands become narrower. The imaginary part of conductivity
also decreases for larger $g$.

Figure 6: Transmission spectra of devices based on a nitrogen-carbon quasi-
fractal nanoribbon for generations $g=1$ (a), 2 (b), and 3 (c). Figure 7:
Transfer eigenstates in quasi-fractal nitrogen-carbon nanoribbons. The states
correspond to points A, B, C, and D on the transmission spectra of the quasi-
fractal nanoribbon for generations $g=1$ (a), 2 (b), and 3 (c,d).

In quasi-fractal nanoribbons, we also observe a transition from transmission
bands to narrow transmission channels (Fig. 6). When calculating the
transmission spectra, nanoribbons were chosen with the same length equal to
$L=109.5$ Å, but with a different number of quasi-fractal units. In
QuantumATK, the transfer matrix is calculated according to the following
formula

$T_{nm}(E,\mathbf{k})=\sum_{\ell}t_{n\ell}(E,\mathbf{k})t^{\dagger}_{\ell
m}(E,\mathbf{k}),$

where $t_{nk}$ is the amplitude of the transfer from the Bloch state
$\psi_{n}$ in the left electrode to the Bloch state $\psi_{k}$ in the right
electrode, the matrix $t^{\dagger}$ is an Hermitian conjugate. The
transmittance is defined as the trace of the transmission matrix,

$T(E,\mathbf{k})=\sum_{n}T_{nn}(E,\mathbf{k}).$

Let $\lambda_{\alpha}$ be the eigenvalues of the transfer matrix $T_{nm}$.
From the invariance of the trace of the matrix:

$T(E,\mathbf{k})=\sum_{\alpha}\lambda_{\alpha}(E,\mathbf{k}),$

where $\lambda_{\alpha}\in[0,1]$ are transmission eigenvalues for each spin
channel.

The transmission eigenstates are calculated by diagonalizing a linear
combination of Bloch states, $\sum_{n}e_{\alpha,n}\psi_{n}$, where
$e_{\alpha,n}$ are vectors of the basis diagonalizing the transmission matrix:

$\sum_{m}T_{nm}e_{\alpha,m}=\lambda_{\alpha}e_{\alpha,n}.$

Fig. 6 shows the transmission spectra for devices based on a nitrogen-carbon
quasi-fractal nanoribbon for generations $g=1$, 2 and 3. Fig. 7 demonstrates
examples of transfer eigenstates in quasi-fractal nitrogen-carbon nanoribbons.
These states correspond to the peaks in the transmission spectra marked with
points A, B, C and D in Fig. 6. The transmission eigenstates are distributed
throughout the structure and take on maximum values at the edges of the quasi-
fractal structure.

##  3 Thermoelectric properties

In this section, we estimate thermoelectric properties of quasi-fractal
nanoribbons. The search for new thermoelectric nanosystems is an important
area of research associated with possible applications in power generation and
cooling systems on nanoscales. Several studies point to the important role of
quasi-one-dimensional geometry in improving the thermoelectric properties of
[15, 16] nanosystems.

The maximum efficiency of the energy conversion process in a thermoelectric
material is determined by its thermoelectric figure of merit $ZT$, determined
by the expression

$ZT=\frac{S^{2}GT}{\lambda},$

where $S$ is the Seebeck coefficient, $G$ electrical conductivity, $T$
absolute temperature, $\lambda$ thermal conductivity, which is equal to the
sum of the electron $\lambda_{e}$ and phonon $\lambda_{\mathrm{ph}}$ thermal
conductivity.

The transfer coefficients were calculated using the NEGF-method, DFT, and
nonequilibrium molecular dynamics. We used a model in which the central part
(quasi-fractal nanoribbon) is connected to semi-infinite left and right
electrodes (graphene nanoribbons). QuantumATK [14] calculates the specified
thermoelectric coefficients and Peltier coefficient according to the linear
response theory. The following relationships are used

${G_{\rm{e}}}={\left.{{{dI}\over{d{V_{{\rm{bias}}}}}}}\right|_{dT=0}},\
S=-{\left.{{{d{V_{{\rm{bias}}}}}\over{dT}}}\right|_{I=0}},\
{\lambda_{\rm{e}}}={\left.{{{d{I_{Q}}}\over{dT}}}\right|_{I=0}},\
\Pi={\left.{{{{I_{Q}}}\over I}}\right|_{dT=0}}=S{V_{{\rm{bias}}}}.$

Here $I_{Q}=dQ/dT$ is the electronic component of the heat flux.

We use DFT to calculate electron transmission and MD method to calculate
phonon transmission. In the MD method, we use the optimized empirical
potential ReaxFF (2015) [17] for the molecular dynamics of systems containing
carbon C, nitrogen N, hydrogen H, and boron B (the latter component is absent
in our case). Previously, similar computational tools have been successfully
applied to a number of carbon and non-carbon nanoscale systems (see e.g. [18]
for details).

Figure 8: Conductance (a), thermal conductivity (b) and thermoelectric figure
of merit (c) of devices based on quasi-fractal nitrogen-carbon nanoribbons.

Fig. 8 shows conductance, thermal conductivity and thermoelectric figure of
merit of devices based on quasi-fractal nitrogen-carbon nanoribbons with
molecular quasi-fractal blocks of three generations. As can be seen from the
electronic transmission spectra, with an increase in the number of generations
$g$, the conductivity decreases due to the localization of electrons in
fragments of quasi-fractal system. Thermal conductivity due to phonons does
not behave monotonically with a change of $g$:
$\lambda_{\mathrm{ph}}(E;g=1)<\lambda_{\mathrm{ph}}(E;g=3)<\lambda_{\mathrm{ph}}(E;g=2)$.
The obtained values of the thermoelectric figure of merit are several tenths,
and at some values of the energy are close to 1, which indicates the possible
use of the considered nanoribbons for thermoelectric applications, although it
should be noted that with an increase in $g$ the characteristic values of ZT
decrease.

##  4 Conclusion

The method of packaging molecular Sierpinski triangles in one-dimensional
crystals by the method of templates in a super high vacuum allows to create
nanoribbons with quasi-fractal geometry. In this work, using calculations
based on DFT, MD and NEGF methods, we study electronic and phonon transport in
a device based on a quasi-fractal carbon nitride nanoribbon with Sierpinski
triangle blocks.

Usually fractal systems are characterized by a wider absorption frequency
range. Contrary to this, we observe that the absorption bands are constricted
for larger $g$ and the corresponding structures are characterized by a
narrower absorption range. This is due to the fact that, in comparison with
macroscopic fractals, in atomistic structures, along with a change in
geometry, the electronic states of the system are modified. The spectrum of
the quasi-fractal system under consideration tends to go from the spectrum of
the periodic 2d system to the spectrum of an isolated molecular system.

The obtained values of the thermoelectric figure of merit are several tenths,
and at some values of the energy are close to 1, which indicates the possible
use of the considered nanoribbons for thermoelectric applications, although it
should be noted that with an increase in $g$ the characteristic values of ZT
decrease.

## Acknowledgement

The work is partially supported by the Ministry of Science and Higher
Education of the Russian Federation (project 0591-2021-0002).

## References

  * [1] F D M Haldane.  ’Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1d interacting spinless fermi gas.  Journal of Physics C: Solid State Physics, 14(19):2585, 1981.
  * [2] Yuanbo Zhang, Yan-Wen Tan, Horst L Stormer, and Philip Kim.  Experimental observation of the quantum hall effect and Berry’s phase in graphene.  nature, 438(7065):201–204, 2005.
  * [3] Sander N Kempkes, Marlou R Slot, Saoirsé E Freeney, Stephan JM Zevenhuizen, Daniel Vanmaekelbergh, Ingmar Swart, and C Morais Smith.  Design and characterization of electrons in a fractal geometry.  Nature physics, 15(2):127–131, 2019.
  * [4] Na Li, Gaochen Gu, Xue Zhang, Daoliang Song, Yajie Zhang, Boon K Teo, Lian-mao Peng, Shimin Hou, and Yongfeng Wang.  Packing fractal Sierpiński triangles into one-dimensional crystals via a templating method.  Chemical Communications, 53(24):3469–3472, 2017.
  * [5] Chao Li, Xue Zhang, Na Li, Yawei Wang, Jiajia Yang, Gaochen Gu, Yajie Zhang, Shimin Hou, Lianmao Peng, Kai Wu, et al.  Construction of Sierpinski triangles up to the fifth order.  Journal of the American Chemical Society, 139(39):13749–13753, 2017\.
  * [6] Mohammed Ghadiyali and Sajeev Chacko.  Confinement of wave-function in fractal geometry, a detection using DFT.  arXiv preprint arXiv:1904.11862, 2019.
  * [7] Tom Westerhout, Edo van Veen, Mikhail I Katsnelson, and Shengjun Yuan.  Plasmon confinement in fractal quantum systems.  Physical Review B, 97(20):205434, 2018.
  * [8] Thomas Garm Pedersen.  Graphene fractals: Energy gap and spin polarization.  Physical Review B, 101(23):235427, 2020.
  * [9] Alireza Khalili Golmankhaneh and Karmina Kamal Ali.  Fractal Kronig-Penney model involving fractal comb potential.  Journal of Mathematical Modeling, 9(3):331–345, 2021.
  * [10] Xiao-Yun Xu, Xiao-Wei Wang, Dan-Yang Chen, C Morais Smith, and Xian-Min Jin.  Quantum transport in fractal networks.  Nature Photonics, 15(9):703–710, 2021.
  * [11] TS Miller, A Belen Jorge, TM Suter, A Sella, F Corà, and PF McMillan.  Carbon nitrides: synthesis and characterization of a new class of functional materials.  Physical Chemistry Chemical Physics, 19(24):15613–15638, 2017.
  * [12] Asadollah Bafekry, Saber Farjami Shayesteh, and Francois M Peeters.  Two-dimensional carbon nitride (2d-CN) nanosheets: Tuning of novel electronic and magnetic properties by hydrogenation, atom substitution and defect engineering.  Journal of Applied Physics, 126(21):215104, 2019.
  * [13] Asadollah Bafekry, M Faraji, Siavash Karbasizadeh, Nguyen N Hieu, Yee Sin Ang, Abdolhosseini Sarsari, and Mitra Ghergherehchi.  Two-dimensional dirac half-metal in porous carbon nitride C6N7 monolayer via atomic doping.  Nanotechnology, 2021.
  * [14] Søren Smidstrup, Troels Markussen, Pieter Vancraeyveld, Jess Wellendorff, Julian Schneider, Tue Gunst, Brecht Verstichel, Daniele Stradi, Petr A Khomyakov, Ulrik G Vej-Hansen, et al.  QuantumATK: an integrated platform of electronic and atomic-scale modelling tools.  Journal of Physics: Condensed Matter, 32(1):015901, 2019.
  * [15] Yijian Ouyang and Jing Guo.  A theoretical study on thermoelectric properties of graphene nanoribbons.  Applied Physics Letters, 94(26):263107, 2009.
  * [16] Takashi Kodama, Masato Ohnishi, Woosung Park, Takuma Shiga, Joonsuk Park, Takashi Shimada, Hisanori Shinohara, Junichiro Shiomi, and Kenneth E Goodson.  Modulation of thermal and thermoelectric transport in individual carbon nanotubes by fullerene encapsulation.  Nature materials, 16(9):892–897, 2017.
  * [17] Sung Jin Pai, Byung Chul Yeo, and Sang Soo Han.  Development of the reaxff cbn reactive force field for the improved design of liquid CBN hydrogen storage materials.  Physical Chemistry Chemical Physics, 18(3):1818–1827, 2016.
  * [18] Troels Markussen, Antti-Pekka Jauho, and Mads Brandbyge.  Electron and phonon transport in silicon nanowires: Atomistic approach to thermoelectric properties.  Physical Review B, 79(3):035415, 2009.