Investigating exclusive $\rho^{0}$ photoproduction

within the Regge phenomenology approach

László Jenkovszky a⋆, Érison S. Rochab♣ and Magno V. T. Machadob♠

a Bogolyubov ITP, National Academy of Sciences of Ukraine, Kiev 03143 Ukraine

b HEP Phenomenology Group, CEP 91501-970, Porto Alegre, RS, Brazil

###### Abstract

The elastic differential and integrated total cross section for the exclusive
$\rho^{0}$ photoproduction in electron-proton ($ep$) collisions are evaluated
taking into account nonperturbative Pomeron exchange approach. By using three
different models based on Regge phenomenology the results are compared to
recent measurements by H1 Collaboration in $ep$ collisions and by the CMS
collaboration from ultraperipheral proton-lead collisions. The analysis is
expanded by calculating the coherent nuclear cross section, $\sigma(\gamma
A\rightarrow\rho^{0}A)$, which is applied to $\rho^{0}$ production in
ultraperipheral lead-lead and xenon-xenon collisions. The predictions are
compared to the measurements performed by ALICE Collaboration. Aspects of the
theoretical uncertainties and limitations of the formalism are scrutinized.

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##  1 Introduction

The exclusive $\rho^{0}$ photoproduction has been investigated recently both
experimentally and theoretically [1, 2, 3, 4]. Light vector mesons have not a
hardness scale associated to the process in the photoproduction limit.
Therefore they can be used to test the nonperturbative (soft) regime of the
strong interactions. In the context of parton (gluon) saturation approaches
the transition between the region described by perturbative Quantum
Chromodynamics (pQCD) and the soft regime is interpreted in terms of the
saturation momentum scale [5, 6], $Q_{s}(x)$ ($x$ is the Bjorken invariant).
In the color dipole picture [7, 8] the saturation scale characterizes the
boundary on the maximum phase-space gluon density to be reached in the
wavefunction of the hadron. In principle, the light meson photoproduction
dynamics at high energies would be driven by $Q_{s}$ as it reaches values
$\lesssim 1$ GeV in this regime. The perturbative description is even better
for the scattering on nuclei targets as the nuclear saturation scale,
$Q_{s,A}^{2}\propto A^{1/3}Q_{s}^{2}$, is enhanced. Nevertheless, it is
already known that such an approach is unable to describe correctly the total
photoproduction cross section and $\rho^{0}$ production at $Q^{2}=0$ VeV2 [10,
11]. In particular, nonperturbative corrections to the description are needed
and usually they are introduced in the photon wavefunction for color dipoles
of large transverse size.

On the other hand, Regge phenomenology is historically appropriated formalism
to describe soft and diffractive processes, Deeply Virtual Compton Scattering
(DVCS) as well as light meson photoproduction. The production amplitude is
written in a Regge-factorized structure with the corresponding coupling of
particles to the Pomeron. The introduction of a hard scale dependence suitable
for vector meson electroproduction can be carefully constructed based on
geometric arguments. The Reggeometric Pomeron model [12, 13] is one example of
such a class of phenomenological formalism. The approach does a good job in
describing the vector meson photo and electroproduction at the DESY-HERA
energy regime considering a nucleon target. The possibility for testing these
models in the coherent vector meson production in ultraperipheral heavy ion
collisions (UPCs) is now a reality. The basic idea is that the production
cross section in $AA$ collisions can be factorized in terms of the equivalent
flux of photons of the colliding nucleus and the photon-target production
cross section [14].

In this work the $\rho^{0}$ photoproduction in $ep$ and $AA$ collisions will
be examined closely. Three different models based on Regge phenomenology [15]
are investigated: the Reggeometric Pomeron (RP) model [12, 13], the Soft
Dipole Pomeron (SDP) [16, 17, 18] and a factorized Regge model with nonlinear
Pomeron trajectory (NL) [19]. In $ep$ collisions the elastic differential
cross section and the integrated total cross section will be analysed. The
results are compared to recent measurements performed by HERA-H1 [1] and CMS
collaborations [2]. Making used of Vector Dominance Model (VDM) and Glauber
multiple scattering formalism the coherent cross section is computed. The
results are compared to the measurements for $\rho^{0}$ production in PbPb and
XeXe UPCs done by ALICE Collaboration at the Large Hadron Collider (LHC) [3,
4]. Namely, the rapidity distribution has been determined in PbPb UPCs at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV and XeXe UPCs at
$\sqrt{s_{\mathrm{NN}}}=5.44$ TeV. The present investigation extends our
earlier works [20, 21, 22, 23, 24] where the rapidity distribution and
$|t|$-dependence of quarkonium production in proton-proton and UPCs in $AA$
collisions have been addressed. The investigation is also relevant for the
physics of the Electron Ion Collider (EIC) [25], Electron-ion collider in
China (EicC) [26] and the Large Hadron Electron Collider (LHeC) [27]. The
topic is matter of intense theoretical studies in recent years [28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39].

This paper is organized as follows. In Sec. 2 we briefly review the exclusive
$\rho^{0}$ meson production, $\gamma+p\rightarrow\rho^{0}+p$, in the context
of the Reggeometric, Soft Dipole and nonlinear trajectory Pomeron models.
Afterwards, using VDM approach and Glauber formalism for nuclear effects, the
expressions for the coherent nuclear cross section is revised. In section 3
the theoretical calculation is compared to available experimental measurements
in $ep$ and $AA$ UPCs collisions. is predicted and applied to UPCs at the LHC
energies. A discussion on the theoretical uncertainties is done. In the last
section the main results are summarized.

Figure 1: Diagrams of DVCS (a) and vector meson production (b) in lepton-
nucleon scattering; (c) DVCS/meson production amplitude in a Regge-factorized
structure including representation for particle vertices, $V_{1}(Q^{2},t)$ and
$V_{2}(t)$.

##  2 Theoretical framework

Exclusive $\rho^{0}$ photoproduction process, $\gamma+p\rightarrow\rho^{0}+p$,
will be described by using three models based on Regge phenomenology (see Fig.
1-b). The first one is the Reggeometric Pomeron (RP) model, the second is the
Soft Dipole Pomeron (SDP) model and the third one is a factorized Regge model
with nonlinear pomeron trajectory (NL). All of them are also able to describe
electroproduction data. In general case, the hardness parameter is given by
$\widetilde{Q}^{2}=Q^{2}+M_{V}^{2}$, where $Q^{2}$ is the photon virtuality
and $M_{V}$ is the vector meson mass.

The elastic differential cross section, $d\sigma_{el}/dt$, and the total cross
section, $\sigma(\gamma^{*}+p\rightarrow\rho^{0}+p)$, associated to the
single-component RP model in a given scale $\widetilde{Q}^{2}$ are given by
[12, 13]:

$\displaystyle\frac{d\sigma_{el}^{\mathrm{RP}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{\left(1+\frac{\widetilde{Q^{2}}}{{Q_{0}^{2}}}\right)^{2n}}\left(\frac{W_{\gamma
p}^{2}}{W_{0}^{2}}\right)^{2(\alpha(t)-1)}\exp\left[B_{0}(\widetilde{Q^{2}})\,t\right],\quad
B_{0}(\widetilde{Q^{2}})=4\left(\frac{a}{\widetilde{Q^{2}}}+\frac{b}{2m_{N}^{2}}\right),$
(1) $\displaystyle\sigma_{\gamma p\to Vp}^{\mathrm{RP}}$ $\displaystyle=$
$\displaystyle\frac{A_{0}^{2}}{\left(1+\frac{\widetilde{Q}^{2}}{\widetilde{Q}^{2}_{0}}\right)^{2n}}\frac{\left(W_{\gamma
p}/W_{0}\right)^{4(\alpha_{0}-1)}}{B\left(W_{\gamma
p},\widetilde{Q}^{2}\right)},\quad B\left(W_{\gamma
p},\widetilde{Q}^{2}\right)=B_{0}(\widetilde{Q^{2}})+4\alpha^{\prime}\ln\left(\frac{W_{\gamma
p}}{W_{0}}\right),$ (2)

where the quantity $B_{0}(\widetilde{Q^{2}})$ reflects the geometrical nature
of the model and $\alpha(t)$ denotes the effective Pomeron trajectory. The
first and second term in $B_{0}(\widetilde{Q^{2}})$ correspond to the
effective sizes of upper and lower vertices in Fig. 1-c, respectively. In the
formulas above, $W_{0}=1$ GeV.

Table 1: Values of the parameters for the Reggeometric Pomeron model [13] fitted to data on $\rho^{0}$ production measured by DESY-HERA [40, 41, 42]. The recent high precision data of Ref. [1] are not included. $A_{0}$ $\left[\frac{\sqrt{\text{nb}}}{\text{GeV}}\right]$ |  $\widetilde{Q^{2}_{0}}$ $\left[\text{GeV}^{2}\right]$ | $n$ | $\alpha_{0}$ |  $\alpha^{\prime}$ $\left[\text{GeV}^{-2}\right]$ | $a$ | $b$
---|---|---|---|---|---|---
344 $\pm$ 376 |  0.29 $\pm$ 0.14 |  1.24 $\pm$ 0.07 |  1.16 $\pm$ 0.14 |  0.21 $\pm$ 0.53 |  0.60 $\pm$ 0.33 |  0.9 $\pm$ 4.3

In the photoproduction limit $\tilde{Q}^{2}=M_{V}^{2}$. The original
parameters of the model are presented in Table 1, which were determined in
Ref. [13] by using previous DESY-HERA measurements [40, 41, 42]. It should be
noticed that the quality of fit $\chi^{2}/\mathrm{dof}=2.74$ for the
$\rho^{0}$ meson is not optimal and authors called attention that a single
$e^{Bt}$ parametrization is not sufficient to reproduce the $d\sigma/dt$ above
$|t|>0.5$ in both electroproduction and photoproduction regions. It is
symptomatic the large uncertainty on the parameters around the central value
and the significantly large value of the effective pomeron intercept,
$\alpha_{0}=1.16$.

Table 2: Values of the parameters for the Soft Dipole Pomeron model [16] fitted to data on $\rho^{0}$ production at HERA [40, 41, 42].  $A_{1}$ $\left[\frac{\sqrt{\mu\text{b}}}{\text{GeV}}\right]$ |  $A_{2}$ $\left[\frac{\sqrt{\mu\text{b}}}{\text{GeV}}\right]$ |  $b_{1}$ [GeV-2] |  $b_{2}$ [GeV-2] |  $s_{0}$ $\left[\text{GeV}\right]$
---|---|---|---|---
-13.944 | 8.3626 | 3.89 | 1.83 | 1.00

Now, the differential cross section and total cross section for the SDP model
is presented [16, 17, 18]. These observables are expressed as [16]:

$\displaystyle\frac{d\sigma_{el}^{\mathrm{SDP}}}{dt}$ $\displaystyle=$
$\displaystyle\left(\frac{s}{s_{0}}\right)^{2\alpha(t)-2}\left\\{\left[G_{1}(t)+G_{2}(t)\ln{\left(\frac{s}{s_{0}}\right)}\right]^{2}+\frac{\pi^{2}}{4}G_{2}^{2}(t)\right\\},$
(3) $\displaystyle\sigma_{\gamma p\to Vp}^{\mathrm{SDP}}$ $\displaystyle=$
$\displaystyle
4\pi\left\\{\frac{A_{1}^{2}}{2B_{1}}+2\frac{A_{1}A_{2}}{B_{1}+B_{2}}\ln\left(\frac{W_{\gamma
p}^{2}}{W_{0}^{2}}\right)+\frac{A_{2}^{2}}{2B_{2}}\left[\ln^{2}\left(\frac{W_{\gamma
p}^{2}}{W_{0}^{2}}\right)+\frac{\pi^{2}}{4}\right]\right\\},$ (4)

where the residua of the single, $G_{1}$, and double pole, $G_{2}$, are
parametrized as follows,

$\displaystyle G_{1}(t)=A_{1}e^{b_{1}t}g_{1}(t),\quad
G_{2}(t)=A_{2}e^{b_{2}t}g_{2}(t).$ (5)

In the phenomenology, $G_{1}(t)$ factorizes into a usual $pI\\!\\!Pp$-vertex,
$G_{1}\propto e^{b_{1}t}$, and a $\gamma I\\!\\!PV$-vertex, $g_{1}(t)$, which
in general is parametrized by a polynomial on $t$. The residue of the double
pole $G_{2}$ can be cast from $G_{1}$, but such a correlation can be relaxed.
For instance, in Ref. [17] one uses $b_{1}=b_{2}\sim 2.25$ GeV-2,
$g_{1}=(1+h_{1}t)$ and $g_{2}\approx(1+h_{2}t)$. In order to obtain Eq. (4)
the simplest option has been considered following Ref. [16], i.e.
$g_{1}(t)=g_{2}(t)=1$. This is enough for the present study. Here, a linear
trajectory for the Pomeron, $\alpha(t)=1+\alpha^{\prime}t$ is considered, with
$\alpha^{\prime}=0.25$ GeV-2. Moreover, in Eq. (4) the following notation is
employed, $B_{i}=b_{i}+\alpha^{\prime}\ln(W_{\gamma p}^{2}/W_{0}^{2})$, with
$i=1,2$ and $W_{0}=1$ GeV.

The parameters in Ref. [16] are fitted including the single pomeron exchange,
the $f_{2}$ contribution and the $I\\!\\!P\\!-\\!f_{2}$ interference. Here,
only the $I\\!\\!P$-exchange is taken into account and the parameters
$A_{1,2}$ were tuned to describe the old DESY-HERA data [40, 41, 42] whereas
$b_{1,2}$ are fixed as their original values ($b_{1}=3.89$ GeV-2 and
$b_{2}=1.83$ GeV-2 ). The values of the parameters for $\rho^{0}$ are
presented in Table 2.

Finally, a Regge model taking into account nonlinear Pomeron trajectory (NL)
[19] is investigated. It is a factorized Regge pole model where the dependence
on the mass and virtuality of the external particles enters via the relevant
residue functions. On the other hand, the Pomeron trajectory is universal and
scale-independent with the form
$\alpha(t)=\alpha_{0}-\alpha_{1}\ln(1-\alpha_{2}t)$. Here, the $\alpha_{i}$
are the $\alpha(t)$-trajectory parameters. It produces a approximately linear
behaviour at small $|t|$, with $\alpha^{\prime}=\alpha_{1}\alpha_{2}$ being
the forward slope. Moreover, at large $|t|$ the cross section presents scaling
behaviour driven by the quark counting rule [19]. The invariant scattering
amplitude and the differential cross section are written as:

$\displaystyle A(s,t,\widetilde{Q}^{2})_{\gamma^{*}p\rightarrow
Vp}^{\mathrm{NL}}=-A_{0}\,4\pi\,V_{1}(t,\widetilde{Q}^{2})\,V_{2}(t)\,\left(\frac{-is}{s_{0}}\right)^{\alpha(t)-1},\quad\frac{d\sigma_{el}^{\mathrm{NL}}}{dt}=\frac{1}{16\pi}\left|A(s,t,\widetilde{Q}^{2})\right|^{2},$
(6)

where $A_{0}$ is the normalization factor,
$V_{1}(t,\widetilde{Q}^{2})=\exp[b_{2}\beta(z)]$ is the $\gamma
I\\!\\!PV$-vertex and $V_{2}(t)=$ exp$[b_{1}\alpha(t)]$ is the
$pI\\!\\!Pp$-vertex. The quantities $\beta(z)$ and $\alpha(t)$ are the
exchanged Pomeron trajectory in the meson vertex and in the proton vertex,
respectively. In Ref. [19] the vertex $V_{1}$ has been modeled in such way
that it contains $t$-dependence and also $Q^{2}$-dependence through the
trajectory, $\beta(z)=\beta_{0}-\beta_{1}\ln(1-\beta_{2}z)$, where
$z=t-\widetilde{Q}^{2}$. The constants $\beta_{i}$ are the
$\beta(z)$-trajectory parameters. Accordingly,
$\beta_{1}\beta_{2}=\beta^{\prime}$ is the forward slope of this trajectory.
In exclusive vector meson production, the hardness in this model is the same
as for the Reggeometric Pomeron one, $\widetilde{Q}^{2}=Q^{2}+M_{V}^{2}$.

The calculation of the integrated cross section with the nonlinear trajectory
from the amplitude in Eq. (6) produces the following expression in the limit
$s\rightarrow\infty$ and negligible nucleon mass [19]:

$\displaystyle\sigma(s,\tilde{Q}^{2})^{\mathrm{NL}}_{\gamma^{*}p\rightarrow
Vp}=\int^{\infty}_{0}~{}d|t|~{}\frac{d\sigma(s,t,\tilde{Q}^{2})}{dt}=\frac{K}{\mu-1}{~{}_{2}}F_{1}\left(\nu,\mu-1;\mu;1-\beta\right),$
(7)

where ${}_{~{}2}F_{1}\left(\nu,\mu-1;\mu;1-\beta\right)$ is the Gauss
hypergeometric function. The remaining quantities appearing in the equation
above are defined by:

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{\pi|A_{0}|^{2}}{\alpha_{2}}e^{2\alpha_{0}(b_{1}+b_{2})}(s/s_{0})^{2\alpha_{0}-2},\quad\mu=2\alpha_{1}\left[b_{1}+b_{2}+\ln(s/s_{0})\right]$
(8) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle
1+\alpha_{2}\tilde{Q}^{2},\quad\nu=2b_{2}\alpha_{1}.$ (9)

The fixed parameters of the model are $\alpha_{0}=\beta_{0}=1.09$,
$\alpha_{1}=\beta_{1}=2.00$, $s_{0}=1$ GeV2, $\alpha_{2}=\beta_{2}=0.125$
GeV-2 and $b_{1}=2.00$, which are constrained by plausible assumptions [19].
There are only two free parameters: the parameter $b_{2}$, entering the
$\gamma I\\!\\!PV$-vertex and the normalization factor, $|A_{0}|^{2}$. In the
the original paper [19], they were only fitted to the DESY-HERA
electroproduction data. Here, we have tuned them to describe the $\rho^{0}$
photoproduction data measured in Refs. [40, 41, 42]. One obtains the values,
$b_{2}=2.300$ and $|A_{0}|^{2}=0.885$ nb. It was checked that the new tuned
values are consistent with the original ones, for instance $b_{2}=1.087$ and
$|A_{0}|^{2}=0.839$ nb at $Q^{3}=3.3$ GeV2. Of course, the factor two of
enhancement on $b_{2}$ parameter will bring consequences for the $t$-slope in
the photoproduction limit.

We are also interested in the nuclear coherent production of $\rho^{0}$, which
is relevant for the physics at the EIC, EicC and LHeC [25, 26, 27]. Moveover,
the coherent production is the main input for vector meson production in
Ultraperipheral Heavy Ion Collisions (UPCs). The nuclear effects for the
process, $\gamma+A\rightarrow\rho^{0}+A$ are described here by vector
dominance model (VDM) and Glauber approach. Namely, using the classical
mechanics Glauber formula for multiple scattering of the $\rho^{0}$ in the
nuclear medium [43] the differential cross section at $t=0$ is given by:

$\displaystyle\left.\frac{\mathrm{d}\sigma\left(\gamma+A\to\rho^{0}+A\right)}{\mathrm{d}t}\right|_{t=0}=\frac{\alpha_{em}}{4f^{2}_{\rho}}\sigma_{tot}^{2}\left(\rho^{0}A\right)=\frac{\alpha_{em}}{4f^{2}_{\rho}}\left\\{\int\mathrm{d}^{2}\textbf{b}\left[1-\exp\left(-\sigma_{tot}\left(\rho^{0}p\right)T_{A}\left(\textbf{b}\right)\right)\right]\right\\}^{2},$
(10)

where $T_{A}(b)$ is the nuclear thickness function and $f_{\rho}$ is the
vector-meson coupling. The value $f^{2}_{\rho}/4\pi=2.02$ is considered in
calculations. The formalism above has been extended in the generalized vector
dominance model [44] (or Gribov-Glauber model), which takes into account
coherence effects for the soft interactions for production of $\rho$ and
$\rho^{\prime}$-mesons. The input for the Glauber model calculation in Eq.
(10) is the cross section for the process $\rho^{0}+p\rightarrow\rho^{0}+p$,
which is given by:

$\displaystyle\sigma_{tot}\left(\rho^{0}p\right)=\sqrt{\frac{4f^{2}_{\rho}}{\alpha_{em}}\left.\frac{\mathrm{d}\sigma\left(\gamma+p\to\rho^{0}+p\right)}{\mathrm{d}t}\right|_{t=0}},$
(11)

where the predictions from the three phenomenological models will be
introduced in Eq. (11) . The corresponding integrated cross section is
expressed as:

$\displaystyle\sigma(\gamma+A\rightarrow\rho^{0}+A)=\left.\frac{d\sigma(\gamma+A\rightarrow\rho^{0}+A)}{dt}\right|_{t=0}\int\limits_{t_{min}}^{\infty}\mathrm{d}|t|\,\left|F_{A}\left(t\right)\right|^{2},$
(12)

where $F_{A}$ is the nuclear form factor. An analytic form factor given by a
hard sphere of radius, $R_{A}=r_{0}A^{1/3}$ fm, convoluted with a Yukawa
potential with range $a$ [45] will be considered. This is sufficient for
practical utilization as shown in Ref. [46]. The analytical expression is the
following,

$\displaystyle
F_{A}(|q|)=\frac{4\pi\rho_{0}}{A|q^{3}|}\left(\frac{1}{1+a^{2}q^{2}}\right)\left[\sin{(|q|R_{A})}-|q|R_{A}\cos{(|q|R_{A})}\right],$
(13)

where $A$ is the mass number, $q$ is the momentum transfer, $\rho_{0}=3/(4\pi
r_{0})$ fm-3 and $a=0.7$ fm.

In next section we investigate the $\rho^{0}$ photoproduction on nucleons and
the nuclear coherent scattering by using as input the Regge models discussed
before. The prediction will be compared to updated data sets. As a by product,
the coherent production of $\rho^{0}$ at UPCs is analysed for both PbPb and
XeXe collisions at the LHC.

Figure 2: Cross section of exclusive $rho^{0}$ meson photoproduction as a
function of centre of mass energy of the photon-nucleon system, $W_{\gamma
p}$. Experimental measurements of H1, ZEUS [1, 40, 41, 42] and CMS [2] are
presented. Reggeometric Pomeron prediction is represented by dashed curve, the
Soft Dipole by solid curve and that one for Nonlinear Trajectory by the dot-
dashed curve.

##  3 Results and discussions

Let us start with the numerical results and their comparison to recent
experimental measurements for $\rho^{0}$ photoproduction by H1-HERA [1] and
CMS Collaboration [2] at the LHC. In the last case, the the points has been
extracted from ultraperipheral pPb coherent collisions at 5.02 TeV.

First, in Fig. 2 the cross section of $\rho$ production in $\gamma p$
scattering is shown as a function of the photon-nucleon centre of mass energy.
Previous measurements of the photoproduction cross section determined by H1
and ZEUS are also presented. The Reggeometric Pomeron model is labeled by the
dashed-line, the Soft Dipole model by solid line and Nonlinear Pomeron
trajectory by dot-dashed line. It should be stressed that the models
parameters have been determined by fitting the old H1 and ZEUS data [40, 41,
42]. Here, a parameter free calculation is performed. The Reggeometric Pomeron
model has the steeper energy dependence and the others presenting a milder
growth. This is traced back to the effective pomeron intercept in the former,
$\alpha_{0}=1.16$ whereas in the Soft Dipole model the energy growth is
proportional to $\ln^{2}(W_{\gamma p}^{2}/W_{0}^{2})$. In the nonlinear
trajectory model a lower pomeron intercept is considered, namely
$\alpha_{0}=1.06$ compared to the Reggeometric one. As expected, once the
parameters have been tuned to old HERA data, the previous measurements by H1
and ZEUS are better described. Given the relatively large error bars in CMS
data, most of the models describe those measurements except the Reggeometric
one at higher energies. It is worth mentioning that the $f_{2}$ contribution
and its interference with the pomeron was not considered here. However, these
contributions can be important at the lower energy data points measured
recently by H1 [1] .

Figure 3: Differential cross section, $d\sigma/dt$, for exclusive $\rho^{0}$
photoproduction as a function of $|t|$ at fixed average energy $\langle
W_{\gamma p}\rangle$ = 50 GeV measured by H1 Collaboration [1]. Same notation
as previous figure.

Secondly, the $|t|$-distribution is investigated and presented in Figs. 3 and
4. The differential cross section is compared to the H1 data [1] in Fig. 3 at
the average energy of $\langle W_{\gamma p}\rangle=50$ GeV as a representative
sample. As expected, both the Reggeometric and Soft Dipole models are able to
describe the overall normalization and the $|t|$ slope at very small-$t$. At
larger $t$ the description loses consistency and underestimate the central
points at larger $t$. On the other hand, the present version of the nonlinear
trajectory model underestimates the small-$t$ and overestimates the large $t$
data points. However, this can be just a consequence of the parameters tuning
as discussed in previous section. It is urged a new high quality fits which
include these new data for $\rho^{0}$ photoproduction in the analysis.

Figure 4: Differential cross section, $d\sigma/dt$, for exclusive $\rho^{0}$
photoproduction as a function of $|t|$ at fixed average energies of $\langle
W_{\gamma p}\rangle$ = 35.6, 59.2, 108.6 and 176.0 GeV measured by CMS
Collaboration [2]. Same notation as previous figure.

Third, in Fig. 4 the models are compared to the CMS Collaboration data [2].
The differential cross section has been measured in the interval $0.025<|t|<1$
GeV2 and for average values of energy, $\langle W_{\gamma
p}\rangle=35.6,\,59.2,\,92.6,\,108.0$ and $176.0$ GeV, respectively.
Concerning the Reggeometric and Soft Dipole models, both underestimate large
$t$ data following the observed behavior against H1 data. Now, the NL model
does a better job compared to the H1 data description. The CMS measurements
cover a wider interval of $t$ mostly including larger values compared to the
H1 ones. It has been reported in Ref. [2] that the STARLIGHT Monte Carlo
prediction is widely higher than the CMS data in the high-$t$ region and the
deviation consistently increase with energy.

At this point, the behavior can be traced back to the average slope coming out
from each model. As an exercise, let us consider $\langle W_{\gamma
p}\rangle=$ 92.6 GeV and compute the predicted $t$-slope by using the original
parameters of the models. In the RP model, the slope is given by Eq. (2) and
$B_{\mathrm{RP}}\left(\langle W_{\gamma
p}\rangle,\widetilde{Q}=m_{\rho}\right)\approx 9.83$ GeV-2. A similar compact
formula may not exist for SDP model due to the non-exponential (power) terms
in the relevant differential cross section (see, for instance Ref. [17] for a
discussion on the slope in SDP model). Concerning the NL model the slope of
the forward cone is,

$\displaystyle
B(s,Q^{2},t)=\frac{d}{dt}\ln|A|^{2}=2\left[b_{1}+\ln\left({\frac{s}{s_{0}}}\right)\right]\frac{\alpha^{\prime}}{1-\alpha_{2}t}+2b_{2}\,\frac{\alpha^{\prime}}{1-\alpha_{2}z},$
(14)

which, in the forward limit, $t=0$ reduces to

$\displaystyle
B(s,Q^{2})=2\left[b_{1}+\ln\left(\frac{s}{s_{0}}\right)\right]\alpha^{\prime}+2b_{2}\,\frac{\alpha^{\prime}}{1+\alpha_{2}\widetilde{Q}^{2}},$
(15)

where $s=W_{\gamma p}^{2}$. The slope $B_{NL}$ shows shrinkage in $s$ and
antishrinkage in $\widetilde{Q}^{2}$. Numerically, one has
$B_{\mathrm{NL}}\left(\langle W_{\gamma
p}\rangle,\widetilde{Q}=m_{\rho}\right)\approx 6.60$ GeV-2

The CMS Collaboration has parametrized the $t$-dependence using the simple
Regge formula, $B(W_{\gamma p})=B_{0}+4\alpha^{\prime}\ln(W_{\gamma
p}/W_{0})$, with $W_{0}=92.6$ GeV, $\alpha^{\prime}=0.28\pm 0.16$ GeV-2 and
$B_{0}\approx 10.3$ GeV-2. data. On the other hand, H1 parametrized in the
same way, with $W_{0}=40$ GeV, $\alpha^{\prime}=0.233\pm
0.064(\mathrm{stat.})$ GeV-2 and $B_{0}=9.59\pm 0.14\,(\mathrm{stat.})$ GeV-2.
Therefore, $B(92.6\,GeV)\approx 10.3$ GeV-2 for CMS and 10.5 GeV-2 for H1.

Figure 5: Nuclear coherent cross section as a function of centre-of-mass
energy of the photon-nucleon system, $W_{\gamma N}$. Reggeometric Pomeron
prediction is represented by dashed curve, the Soft Dipole by solid curve and
that one for Nonlinear Trajectory by the dot-dashed curve. As an illustration,
the dotted curve represents the low energy contribution.

Moving now to the $\rho^{0}$ production on nuclei, predictions to the coherent
nuclear scattering cross section, $\sigma(\gamma+A\rightarrow\rho^{0}+A)$, are
presented in Fig. 5 as a function of photon-nucleon energy, $W_{\gamma N}$. A
comparison is done with the extracted values of the coherent cross sections
performed in Ref. [47] (labeled FGSZ) and by ALICE Collaboration in Ref. [4]
using the measured data on UPCs at the LHC (PbPb collisions at 5.02 TeV). The
low energy cross sections for $\gamma Au$ collisions (FGSZ) is also presented
for completeness (RHIC AuAu collisions). The description is quite reasonable
for the energy dependence and all models underestimate the extracted cross
section by 30%. The RP and SDP models produce almost identical results and the
NL model gives a smaller normalization. It is clear that the similar results
are due to the strong nuclear effects given that the calculated $\sigma(\gamma
p\rightarrow\rho^{0}p)$ shown differences between models at large and low
energies. As an illustration, at low energy we consider a black disk scaling
following Ref. [46], $\sigma(\gamma A\rightarrow\rho^{0}A)\approx
A^{4/3}\sigma(\gamma p\rightarrow\rho^{0}p)=A^{4/3}\,(YW_{\gamma N}^{-\eta})$
(with $Y=26$ $\mu$b and $\eta=1.23$), which is represented by the dotted
curve. In the calculations we added the reggeon contribution to the
photoproduction off nucleons. The $\gamma Xe$ cross section has been extracted
by ALICE Collaboration in Ref. [4] from the measured $d\sigma/dy=131.50\pm
24.96$ (errors summed into quadrature) in XeXe UPC collisions at 5.44 TeV. The
cross section corresponds to the energy of photon-nucleon system $W_{\gamma
N}=65$ GeV and $\sigma(\gamma Xe\rightarrow\rho^{0}Xe)=1.12\pm 0.21$ mb. The
predicted values from the models are 1.07 mb (RP), 1.08 mb (SDP) and 0.92 mb
(NL), respectively.

Figure 6: Rapidity distribution for the coherent photoproduction of $\rho^{0}$
mesons in PbPb UPC in midrapidity at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV .
Predictions for forward-neutron selection (top left), as well as for 0n0n (top
right), 0nXn (bottom left) and XnXn (bottom right) neutron class selections
are presented. Data from ALICE Collaboration [3].

The predictions for the nuclear coherent cross section presented in Fig. 5 and
calculated by using Eq. (12) will be used as input for computing the
$\rho^{0}$ photoproduction is ultraperipheral PbPb and XeXe collisions at the
LHC. The corresponding rapidity distribution in AA UPCs takes a factorized
form in Equivalent Photon Approximation (EPA) approach and it is given by:

$\displaystyle\frac{d^{2}\sigma(A+A\rightarrow
A+\rho^{0}+A)}{dy}=\omega_{+}\frac{dN_{\gamma/A}(\omega_{+})}{d\omega}\sigma_{\gamma
A\rightarrow\rho^{0}A}(\omega_{+})+\omega_{-}\frac{dN_{\gamma/A}(\omega_{-})}{d\omega}\sigma_{\gamma
A\rightarrow\rho^{0}A}(\omega_{-}),$ (16)

where $dN_{\gamma/A}(\omega)/d\omega$ is the photon flux in nucleus $A$ and
$\omega$ is the photon energy. For fixed rapidity $y$ and transverse momentum
$p_{T}^{2}\approx|t|$ of the produced mesons, the photon momentum is given by
$\omega^{\pm}=\frac{m_{\rho}^{2}-t}{2M_{T}e^{\mp}}$. $M_{T}$ is the transverse
mass of the mesons. For simplicity, the analytical expression for the flux of
photons produced by a fast-moving point-like charge has been considered [46,
48].

For large nuclei the process can be followed by additional photon exchanges
between the colliding ions which lead to possible excitation of one or both of
them. This can be accompanied by emission of neutrons moving along the
direction of the ion beams [49]. Usually, the additional photon exchanges can
be accounted by an effective modification of the photon flux. Therefore, a
selection can be done for the different channels, $k$, which are identified by
emission of various number of neutrons. The neutron class selections are for
instance $k=$ 0n0n, 0nXn and XnXn. Thus, the photon flux for channel $k$ is
expressed as [49],

$\displaystyle\frac{dN_{\gamma/A}^{k}(\omega)}{d\omega}=\int_{2R_{A}}^{\infty}d^{2}\vec{b}\,\frac{d^{3}N_{\gamma/A}(\omega,\vec{b})}{d\omega
d^{2}\vec{b}}P_{k}(\vec{b})$ (17)

where $d^{3}N_{\gamma A}/d\omega d^{2}\vec{b}$ is the photon spectrum at a
perpendicular distance $\vec{b}$ from the center of the emitting nucleus. The
quantity $P_{k}$ is the probability to emit a given number of neutrons
corresponding to the channel $k$. Here, we follow the modification of the
photon flux due to the emission of the forward neutrons given in Ref. [49].

Figure 7:  Rapidity distribution for the coherent photoproduction of
$\rho^{0}$ mesons in XeXe UPC in midrapidity at $\sqrt{s_{\mathrm{NN}}}=5.44$
TeV. Predictions of the Reggeometric Pomeron model (dashed curve), Soft Dipole
Pomeron model (solid curve) and nonlinear Pomeron trajectory (dot-dashed
curve) are compared to data from ALICE Collaboration [4].

In Fig. 6 results are shown for coherent nuclear $\rho^{0}$ production for
PbPb peripheral collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV ($p_{T}<0.2$
GeV). The rapidity distribution of the meson is presented and compared to the
ALICE collaboration measurements [3]. As the data refer to the absolute value
of rapidity (filled squares) the data points at positive rapidities have been
reflected into negative rapidities (open squares). In the top panel on the
left the rapidity distribution is shown for no forward-neutron selection. The
remaining plots present the cross section for coherent $\rho^{0}$ production
accompanied by nuclear breakup. Thus, the following forward-neutron classes
selection is shown: 0n0n, 0nXn and XnXn, respectively. The predictions for the
RP model are labeled by the dashed curves, SDP model by the solid curves
whereas the ones for the NL model are labeled by the dotted-dashed curves. The
models underestimate the no forward neutron selection data within the error
bars as expected based on results presented in Fig. 5 . The same occurs for
the 0n0n and 0nXn classes. On the other hand, RP and SDP models describe XnXn
cross section at midrapidity and NL result is close to measurements. Our
calculations produce lower normalizations compared for instance to Guzey,
Kryshen and Zhalov (GKZ) model [47]. It is based on a modified vector-
dominance model where the interaction between the photon hadronic fluctuations
and nucleons is described by the Gribov-Glauber formalism. Moreover, comparing
our calculations to the STARLIGHT Monte Carlo [50] the predictions are
somewhat similar. This can be anticipated as the formalism for nuclear effect
is the same in both cases.

Figure 8:  Rapidity distribution for the coherent photoproduction of
$\rho^{0}$ mesons in PbPb (5.02 TeV, left panel) and XeXe (5.44 TeV, right
panel) UPCs at the LHC. Results for the Reggeometric Pomeron model only.
Comparison of nuclear effects by using CM approach (solid curves) and GG
approach (dashed curves) is done.

In Fig. 7 results are shown for XeXe collisions at the energy of
$\sqrt{s_{\mathrm{NN}}}=5.44$ TeV. The measurement performed by ALICE
Collaboration [4] at midrapidity is also presented. The notation for the
curves is the same as previous figure. In general, the three models are
suitable to predict the magnitude and shape of the rapidity distribution in
XeXe UPCs. The nuclear effects for xenon nucleus are less intense as for lead
nucleus and the cross section normalization now is compatible with data. We
have shown before that the coherent cross section $\sigma(\gamma
Xe\rightarrow\rho^{0}Xe)$ is correctly described by the models discussed here.

Finally, a discussion on the nuclear effects is in order. Our approach is
quite similar to the Starlight one where the calculations are based on the
Regge model for the forward photoproduction cross section on a nucleon target,
vector-meson-dominance (VMD), optical theorem and the Glauber model. The total
cross section of the $\rho^{0}A$ interaction is obtained from classical
mechanics (MC) model, as shown in Eq. (10). On the other hand, the quantum
mechanics expression is given by the Gribov-Glauber (GG) formalism where the
$\rho^{0}A$ cross section is given by:

$\displaystyle\sigma_{tot}^{\mathrm{GG}}(\rho^{0}A)=2\int
d^{2}\vec{b}\left[1-\exp\left(-\frac{1}{2}\sigma_{\rho^{0}N}T_{A}(\vec{b})\right)\right].$
(18)

In the simplification of a sharp sphere nucleus with $\rho_{0}=0.17$ fm-3 and
radius $R_{A}$ one can obtain an estimate of the ratio between the GG and CM
cross sections [51],

$\displaystyle\frac{\sigma_{tot}^{\mathrm{GG}}(\rho^{0}A)}{\sigma_{tot}^{\mathrm{CM}}(\rho^{0}A)}\approx
2\left(1-\frac{3}{2\rho_{0}^{2}\sigma_{\rho^{0}N}^{2}R_{A}^{2}}\right).$ (19)

The ratio is $\approx 1.67$ for lead ($R_{A}\simeq 7.1$ fm) and $1.55$ for
xenon ($R_{A}\simeq 6.1$ fm) by using $\sigma_{\rho^{0}N}\approx 25$ mb. Thus,
the deviation in midrapidity at the LHC energies is very large as one can see
in Fig. 8. In figure, predictions are made by using CM (solide curves) and GG
(dashed curves)) approaches and Reggeometric Pomeron model. Left plot
corresponds to PbPb UPCs and right plot XeXe UPCs at LHC energies.

##  4 Conclusions

In this work predictions for exclusive $\rho^{0}$ photoproduction in $ep$
collisions and also UPCs collisions at the LHC are presented and compared with
the recent experimental measurements. The theoretical approach is based on
Regge phenomenology and 3 different models have been considered: i) the
single-component Reggeometric Pomeron model, ii) the Soft Dipole Pomeron model
and iii) a factorized Regge model with nonlinear Pomeron trajectory. The
comparison of the results to the H1 data has shown that the models are
suitable for the small-$|t|$ region and clear deviations at large $t$. It is
worth mentioning that the models parameters have been tuned to describe
previous measurements by DESY-HERA. Updated fits including new data sets would
be helpful. Concerning the rapidity distributions in PbPb UPCs, RP and SDP
models are in agreement with data in case of no forward neutron selection
whereas NL model underestimates the cross section. In the case of XeXe UPCs
all models are consistent with the data following the same trend observed in
PbPb collisions.

The present paper generalizes results on $\rho$ photoproduction in two
directions. First, we have collected, compared, critically revised and partly
refitting three different Regge-pole models of exclusive $\rho^{0}$
photoproduction. An important conclusion from this comparison is the the
importance of the non-linear nature of Regge trajectories, including that of
the Pomeron. This is seen explicitly e.g. in Fig. 7. Non-linearity of Regge
trajectories is seen both at small $|t|$, manifest as a ”break” in the
differential cross section, of elastic proton-proton scattering as well as at
moderate and large values of $|t|$, matching transition to hard scattering,
described by perturbative QCD. Construction of explicit models of non-linear
Regge trajectories compatible with analytic and asymptotic constrains of the
theory is an important and permanently developing field of research. We intend
to continue research in this direction by including also DIS, DVCS and
production of other vector mesons.

Secondly, we continue investigation of how models of vector meson production
in $ep$ scattering affect the results in ultra-peripheral nuclear collisions.
This direction of research is especially promising also because of the planned
experiments at future accelerators.

## Acknowledgements

We are grateful to Vladyslav Libov for collaborating at an earlier stage of
the work. L.J. was supported by the NASU grant 1230/22-1 Fundamental
Properties of Matter. MVTM was supported by funding agencies CAPES (Finance
Code 001) and CNPq (grant number 306101/2018-1), Brazil.

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