# Exclusive vector meson production with leading neutrons in a saturation
model for the dipole amplitude in mixed space

J. T. Amaral and V. M. Becker  Instituto de Matemática, Estatística e Física –
FURG, Rio Grande, 96203-900 Rio Grande do Sul, Brazil

###### Abstract

We investigate $\rho$ vector meson production in $ep$ collisions at HERA with
leading neutrons in the dipole formalism. The interaction of the dipole and
the pion is described in a mixed-space approach, in which the dipole-pion
scattering amplitude is given by the Marquet-Peschanski-Soyez saturation
model, which is based on the traveling wave solutions of the nonlinear
Balitsky-Kovchegov equation. We estimate the magnitude of the absorption
effects and compare our results with a previous analysis of the same process
in full coordinate space. In contrast with this approach, the present study
leads to absorption $K$ factors in the range of those predicted by previous
theoretical studies on semi-inclusive processes.

##  I Introduction

At high energies, the interaction between the virtual photon and a proton is
described in a convenient way: long before the interaction, the photon splits
into a quark-antiquark pair, or a dipole, and this dipole interacts with the
proton. Thus, in this picture, the virtual photon-proton ($\gamma^{*}p$) cross
section can be written in terms of the dipole-proton cross section, which, by
its turn, is related to the (imaginary part of the) dipole-proton scattering
amplitude. The simplest processes that can be described in terms of the
dipole-proton amplitude are high energy electon-proton ($ep$) collisions,
which were performed at the Hadron Electron Ring Accelerator (HERA). These
collisions can produce a variety of interesting final states and, among them,
those which contain leading neutrons, which are basically neutrons produced at
very small polar angles with respect to the initial proton beam direction,
carrying a large fraction $x_{L}$ of its momentum. These particles can provide
us important insight on strong interactions in the soft regime and, at HERA,
have been first observed in semi-inclusive reactions ($e+p\to e+n+X$, where
$n$ represents the neutron and $X$ is a generic hadronic final state).
Currently, we have access to recent high precision data on leading neutron
production, in both semi-inclusive Aaron2010 ; Olsson2014 ; Andreev2014  and
exclusive reactions Andreev2016 , but already from previous HERA measurements
we have learned that their production are dominated by pion exchange
Adloff1999 ; Chekanov2002 ; Chekanov2004 ; Aktas2005 ; Chekanov2005 ;
Chekanov2007 . Thus, the virtual photon emitted from the incoming electron
interacts with the pion (of the proton cloud) which allows us to extract the
$\gamma^{*}\pi$ cross section. The interaction between the virtual photon and
the pion can also be described in terms of dipole-pion collision, which leads
us to deal with the dipole-pion scattering amplitude. Although our knowledge
on this amplitude is still limited, it can be related to a quantity whose
behavior at high energies is much better known, the dipole-proton scattering
amplitude.

According to the effective theory of color glass condensate (CGC)
JalilianMarian ; PhysRevD.59.014014 ; PhysRevD.59.014015 ; PhysRevD.62.114005
; Iancu2001133 ; Iancu2001145 ; Iancu2001583 ; Ferreiro2002489  for high
energy quantum chromodynamics (QCD), the dipole-proton scattering amplitude,
in the limit of large number of colors ($N_{c}$), is the solution of the
Balitsky-Kovchegov (BK) equation Balitsky ; Kovchegov , which is a nonlinear
equation which gives the evolution of this amplitude with energy. Although
being the simplest nonlinear evolution equation of high energy QCD, BK
equation has no exact solution, but analytical expressions at asymptotical
regimes can be obtained, and these can be used to construct phenomenological
models for the dipole-proton amplitude. One of these models is the so-called
bCGC model Kowalski1 ; Kowalski2 , which is a generalization of the Iancu-
Itakura-Munier model Iancu:2003ge , with the introduction of the dependence on
the impact parameter of the dipole-proton collision. It is important to point
out that bCGC model is a pure coordinate-space model, which means that,
besides depending on the energy, the amplitude depends on the dipole size and
on the impact parameter. This model has been used in the description of the
recent HERA data on leading neutron production in both semi-inclusive
Carvalho:2015eia  and exclusive Goncalves:2015mbf  cases. In the latter case,
the authors applied bCGC model to describe the recent HERA data on exclusive
$\rho$ meson with leading neutron and, besides the good description of the
data, they estimated the contribution of the absorption corrections. These are
related to photon rescattering after interacting with the pion emitted by the
proton and are usually included in an overall constant $K$-factor. Theoretical
studies of semi-inclusive processes predict that the $K$-factor should not be
smaller than $0.7$ Nikolaev:1997cn ; Alesio . For exclusive processes,
experimental and theoretical information on absorption corrections still lack,
but in the coordinate-space phenomenological analysis done in
Goncalves:2015mbf , the values obtained should not be larger than 0.3, meaning
that in exclusive processes the absorption effects would be stronger than in
semi-inclusive ones.

Not only coordinate-space models for the dipole-proton amplitude are available
in the literature. In particular, the saturation model proposed by Marquet,
Peschanski and Soyez (MPS model) Marquet:2007qa  gives an expression for this
amplitude in mixed space, which means that it depends on the collision energy,
the dipole size and, instead of depending on the impact parameter, it depends
on the momentum transferred in the dipole-proton collision. This model is
based on a fundamental property of the BK equation: at asymptotic high
energies, it presents traveling wave solutions Munier , which have been a
natural explanation, for example, to the observed geometric scaling at HERA
Stasto . These solutions were originally obtained in the simplified case where
the amplitude does not depend on the impact parameter (or, equivalently, on
the transferred momentum) Munier , but also found to exist in mixed space and
in full momentum space (when the amplitude depends on the dipole transverse
momentum and the transferred momentum) Marquet1 ; Marquet2 . In this paper we
apply the MPS model to the description of HERA data on exclusive $\rho$ meson
production with leading neutron and compare our results with those obtained in
coordinate-space using bCGC model. As in Goncalves:2015mbf , we assume a
simple relation between the dipole-pion and dipole-proton amplitudes and test
the same models for the virtual pion momentum distribution of the proton.
Among other important differences between both analyses, we find that the
magnitude of the absorption effects are of the same order of that predicted
for semi-inclusive processes, which is in strong contrast to the results
obtained in coordinate space.

This paper is organized as follows. In Sec. II we review the description of
exclusive vector meson production with leading neutron in the dipole
formalism. Section III is devoted to the traveling wave solutions of BK
equation, how these have been generalized to the mixed-space case and the
description of the MPS model. In Sec. IV we apply MPS model to the description
of the HERA data on exclusive $\rho$ meson production with leading neutron,
present our results and compare them to those obtained in full coordinate
space. We summarize our main conclusions in Sec. V.

##  II Exclusive vector meson production with leading neutron in the dipole
model

In the one-pion-exchange approximation OPE , valid for small values of the
(measured) neutron transverse momentum $p_{T,n}$, the $\gamma^{*}p$ cross
section involving leading neutron production can be written as

$\frac{d^{2}\sigma(W,Q^{2},x_{L},t)}{dx_{L}dt}=f_{\pi/p}(x_{L},t)\sigma_{\gamma^{*}\pi}(Y,Q^{2}),$
(1)

where $W^{2}$ is the square of the center-of-mass energy of the virtual
photon-proton system, $Q^{2}$ is the photon virtuality, $Y$ is the total
rapidity interval of the photon-pion system. $x_{L}$, the fraction of the
incoming proton beam energy carried by the leading neutron, and $t$, the four-
momentum transfer squared at the proton vertex, related to the other
kinematical variables by

$t\simeq-\frac{p_{T,n}^{2}}{x_{L}}-\frac{(1-x_{L})(m_{n}^{2}-m_{p}^{2}x_{L})}{x_{L}},$
(2)

where $m_{n}$ ($m_{p}$) is the neutron (proton) mass. The function
$f_{\pi/p}(x_{L},t)$, called pion flux or pion splitting function, describes
the splitting $p\to n\pi^{+}$ and basically gives the virtual pion momentum
distribution in a dressed nucleon (in our case, the proton). There are some
parametrizations $f_{\pi/p}(x_{L},t)$ in the literature; here we will use five
of them, and their explicit forms will be given in Sec. IV.

The interaction of the virtual photon with the pion is given by the cross
section $\sigma_{\gamma^{*}\pi}(Y,Q^{2})$, which is given

$\sigma_{\gamma^{*}\pi}(Y,Q^{2})=\sum_{i=T,L}\int_{-\infty}^{0}\frac{d\sigma_{i}^{\gamma^{*}\pi\to
V\pi}}{dt^{\prime}}dt^{\prime}=\frac{1}{16\pi}\sum_{i=T,L}\int_{-\infty}^{0}\left|{\cal
A}_{i}^{\gamma^{*}\pi\to
V\pi}\right|^{2}\left(1+\beta_{i}^{2}\right)R_{g}^{2}\ dt^{\prime},$ (3)

where ${\cal A}_{i}^{\gamma^{*}\pi\to V\pi}$ is the imaginary part of the
scattering amplitude and the integration is performed over the for-momentum
transfer momentum squared at the pion vertex, $t^{\prime}=-\mathbf{q}^{2}$,
$\mathbf{q}$ denoting the transverse momentum transferred by the pion during
the collision with the virtual photon. The factor
$\left(1+\beta_{i}^{2}\right)$ accounts for the contribution of the real part
of the amplitude, which can be obtained by the dispersion relations Kowalski1

$\beta_{i}=\tan\left(\frac{\pi\lambda}{2}\right),\qquad\lambda=\frac{\partial\log({\cal
A}_{i}^{\gamma^{*}\pi\to V\pi})}{\partial Y},$ (4)

and $R_{g}^{2}$ in (3) incorporates the namely skewness effect (for more
details, see Amir ). $\sigma_{\gamma^{*}\pi}(Y,Q^{2})$ depends on the photon
virtuality and the total rapidity interval of the photon-pion system, given by

$Y=\ln\left(\frac{Q^{2}+\hat{W}^{2}}{Q^{2}+M_{V}^{2}}\right),$ (5)

where $\hat{W}^{2}=W^{2}(1-x_{L})$ the square of the center-of-mass energy of
the $\gamma^{*}\pi$ system and $M_{V}$ is the mass of the produced vector
meson.

At high energies, the process can be seen as a sequence of three factorable
subprocesses, as represented in Fig. 1. The virtual photon with four-momentum
$k$, which in $ep$ collisions is emitted from the incoming electron, splits
into a quark-antiquark pair, or a dipole. This dipole, then, interacts with a
pion $\pi$ of the incident proton $p$ wave function and we have the leading
neutron and the vector meson in the final state.

Figure 1: Exclusive $\gamma^{*}p$ process associated with a leading neutron
$n$ production in the color dipole model.

In this formalism, the amplitude ${\cal A}_{i}^{\gamma^{*}\pi\to V\pi}$ reads

$\displaystyle{\cal A}_{T,L}^{\gamma^{*}\pi\to V\pi}(Y,\mathbf{q})$
$\displaystyle=$ $\displaystyle\int
d^{2}xd^{2}y\int_{0}^{1}dz\Phi_{T,L}^{\gamma^{*}V}(z,\mathbf{x}-\mathbf{y};Q^{2},M_{V}^{2})e^{i\mathbf{q}\cdot\mathbf{y}}T_{\pi}(\mathbf{x},\mathbf{y};Y).$
(6)

where the integration is over the quark and antiquark transverse coordinates,
respectively $\mathbf{x}$ and $\mathbf{y}$, and over $z$, the fraction of
longitudinal momentum carried by the quark of the dipole. The overlap
functions $\Phi_{T,L}^{\gamma^{*}V}(z,\mathbf{x}-\mathbf{y};Q^{2},M_{V}^{2})$
give the probability of splitting of the photon into the vector meson. There
are different models for them in the literature and in this paper we will use
the so-called boosted Gaussian (BG) and light-cone Gaussian (LCG) models
Nemchik ; Frankfurt ; Dosch  (for explicit expressions for the overlap
functions, see Kowalski1 ; Kowalski2 ).

##  III Geometric scaling and the MPS model

In Eq.(6), $T_{\pi}(\mathbf{x},\mathbf{y};Y)$ is the (imaginary part of the)
dipole-pion scattering amplitude. Following Carvalho:2015eia ;
Goncalves:2015mbf  we assume that it can be related to the dipole-proton
scattering amplitude, $T_{p}(\mathbf{x},\mathbf{y};Y)$ through

$T_{\pi}(\mathbf{x},\mathbf{y};Y)=R_{q}T_{p}(\mathbf{x},\mathbf{y};Y),$ (7)

where the factor $R_{q}$ is a constant number, whose value will be discussed
in the next section. $T_{p}(\mathbf{x},\mathbf{y};Y)$ is the solution of BK
equation Balitsky ; Kovchegov , the simplest nonlinear evolution equation of
high energy QCD, which gives the evolution of the dipole-proton amplitude with
rapidity (or, equivalently, with energy). This equation has no exact solution,
but some analytical expressions for the amplitude can be obtained by exploring
the properties of the evolution equation in asymptotical regimes. One of these
properties is that BK equation, at very high energies (i.e., $Y\to\infty$),
admits traveling wave solutions. These (analytical) solutions were obtained
for the first time in Munier , in the simplest case where the dipole amplitude
depends only on the dipole size $r=|\mathbf{r}|=|\mathbf{x}-\mathbf{y}|$ and
the rapidity, i.e. $T_{p}(\mathbf{x},\mathbf{y};Y)=T_{p}(r,Y)$. Besides, the
analysis was done in momentum space, which can be performed in a
straightforward way through a simple Fourier transform. In this space, the
dipole-proton amplitude depends on $k=|\mathbf{k}|$, where $\mathbf{k}$ is the
transverse momentum of the dipole. In coordinate (momentum) space, traveling
wave solutions mean that the amplitude, instead of depending on $r$ ($k$) and
$Y$ separately, depends on them through the scaling variable $rQ_{s}(Y)$
($k/Q_{s}(Y)$), where $Q_{s}(Y)$ is the saturation scale, which separates
different partonic density of the proton target, and grows exponentially with
the rapidity interval, $Q^{2}_{s}(Y)=Q_{0}^{2}e^{\lambda Y}$, with
$\lambda\sim 0.3$ ($Q_{0}$ is a scale related to the target). This behavior of
the dipole amplitude has been a natural explanation for the observation of the
so-called geometric scaling in HERA data Stasto , which, in the case of
inclusive deep inelastic scattering, means that the total cross section
depends on the photon virtuality and energy through the variable
$Q^{2}/Q_{s}^{2}(Y)$.

Traveling wave solutions of BK equation can also be obtained in the case when
the full transverse dependence of the dipole amplitude is considered. This has
been shown in Refs. Marquet1 ; Marquet2 , and a remarkable point is that
geometric scaling is better achieved when the amplitude depends on the dipole
transverse momentum and the momentum transfer of the collision or, in other
words, is in full momentum space, and when the amplitude depends on the dipole
size and the momentum transfer, or in mixed space. The latter case is of
particular interest: after a Fourier transform, one goes from
$T(\mathbf{r},\mathbf{b};Y)$ to $\tilde{T}(\mathbf{r},\mathbf{q};Y)$, where
$\mathbf{q}$ is the momentum transfer (kinematical variable conjugate to the
impact parameter). The resulting mixed-space dipole scattering amplitude
$\tilde{T}(\mathbf{r},\mathbf{q};Y)$ presents the property of geometric
scaling in the regime of small and intermediate values of the momentum
transfer $Q_{0}<|\mathbf{q}|<Q$. In this regime,
$\tilde{T}(\mathbf{r},\mathbf{q},Y)=\tilde{T}(|r|Q_{s}(Y),\mathbf{q})$, with
the saturation scale $Q_{s}^{2}=q^{2}e^{\lambda Y}$. On the other hand, in the
small momentum transfer regime, $|\mathbf{q}|<Q_{0}<Q$, the forward result ,
i.e., $Q^{2}_{s}(Y)=Q_{0}^{2}e^{\lambda Y}$, is recovered.

It is straightforward to express the amplitude (6) in terms of the mixed-space
dipole amplitude. First, through the change of variables
$\mathbf{r}=\mathbf{x}-\mathbf{y}$ and
$\mathbf{b}=z\mathbf{x}+(1-z)\mathbf{y}$ we rewrite it in terms of the dipole
size $\mathbf{r}$ and the impact parameter of the dipole-pion interaction,
$\mathbf{b}$. We get

$\displaystyle{\cal A}_{T,L}^{\gamma^{*}\pi\to V\pi}(Y,\mathbf{q})$
$\displaystyle=$ $\displaystyle\int
d^{2}rd^{2}b\int_{0}^{1}dz\,\Phi_{T,L}^{\gamma^{*}V}(z,\mathbf{r};Q^{2},M_{V}^{2})e^{i\mathbf{q}\cdot(\mathbf{b}-z\mathbf{r})}T_{\pi}(\mathbf{r},\mathbf{b};Y)$
(8)

Then, after introducing the Fourier transform

$\tilde{T}_{\pi}(\mathbf{r},\mathbf{q},Y)=\int d^{2}be^{i\mathbf{q}\cdot
b}T_{\pi}(\mathbf{r},\mathbf{b},Y)$ (9)

we finally obtain

$\frac{d\sigma_{T,L}^{\gamma^{*}\pi\to
V\pi}}{dt^{\prime}}dt^{\prime}=\frac{1}{16\pi}\left|\int
d^{2}r\int_{0}^{1}dz\,\Phi_{T,L}^{\gamma^{*}V}(z,\mathbf{r};Q^{2},M_{V}^{2})e^{-iz\mathbf{q}\cdot\mathbf{r}}\tilde{T}_{\pi}(\mathbf{r},\mathbf{q};Y)\right|^{2},$
(10)

i.e, the cross section for the virtual photon-pion interaction in terms of the
quantity $\tilde{T}_{\pi}(\mathbf{r},\mathbf{q};Y)$, the dipole-pion amplitude
in mixed space.

Based on the extension of geometric scaling to the case of nonzero momentum
transfer, Marquet, Peschanski and Soyez proposed a phenomenological model for
$\tilde{T}_{p}(\mathbf{r},\mathbf{q};Y)$ Marquet:2007qa . It consists in the
following expression for the mixed-space dipole-proton amplitude:

$\tilde{T}_{p}(\mathbf{r},\mathbf{q};Y)=2\pi
R_{p}^{2}f(\mathbf{q})N(rQ_{s}(Y,\mathbf{q})),$ (11)

where $R_{p}$ is the proton radius, $f(\mathbf{q})$ is a form factor, which
catches the transfer dependence of the proton vertex, given by
$f(\mathbf{q})=\exp^{-B\mathbf{q}^{2}}$. The function $N$ is a generalization
of the Iancu, Itakura and Munier (IIM) forward saturation model Iancu:2003ge :

$N(rQ_{s}(Y),Y)=\left\\{\begin{array}[]{ll}N_{0}\frac{rQ_{s}(Y)}{2}^{2\left(\gamma_{c}+\frac{\ln(2/rQ_{s})}{\kappa\lambda
Y}\right)},&rQ_{s}(Y)\leq 2;\\\
1-e^{-a\ln^{2}(brQ_{s}(Y))},&rQ_{s}(Y)>2,\end{array}\right.$ (12)

with a saturation scale depending on the momentum transfer, parametrized as

$Q_{s}^{2}(Y,\mathbf{q})=Q_{0}^{2}(1+c\mathbf{q}^{2})e^{\lambda Y}.$ (13)

This is a simple and intuitive phenomenological model which completely
recovers the geometric scaling in the forward case ($\mathbf{q}=0$) and gives
the saturation scale the correct asymptotic behaviors. Using the MPS model,
Eqs. (11)-(13), and the relation (7) we are able to calculate the cross
section (1) for the exclusive vector meson production with a leading neutron
and confront to available data.

##  IV Results

In this analysis we consider the HERA data on exclusive photoproduction of
$\rho^{0}$ mesons, associated with leading neutrons, presented in Andreev2016
. The data covers the following kinematical ranges: $20<W<100$ GeV and
$Q^{2}<2$ GeV2. The range on the $Q^{2}$ leads to a mean value $\langle
Q^{2}\rangle=0.04$ GeV2. Following Goncalves:2015mbf , here we use $W=60$ GeV
and $Q^{2}=0.04$ GeV2. All the parameters of MPS model are kept fixed. Those
who enter the scaling function $N$ are taken from Ref.Soyez:2007kg , where the
IIM saturation model was extended to include heavy quarks, the values of their
masses being $m_{f}=0.14$ GeV for light flavors, $m_{c}=1.4$ GeV for charm
quark and $m_{b}=4.5$ GeV for bottom quark. The parameters $a$ and $b$ are
uniquely determined from the conditions that $N$ and its derivative are
continuous at $rQ_{s}=4$ and $N_{0}$ is fixed at the value $N_{0}=0.7$. The
saturation scale parameters are $\lambda=0.2197$ and $Q_{0}=0.298$ GeV and the
proton radius is $R_{p}=3.34$ GeV-1. The remaining parameters of MPS model are
$c$ and $B$ and their values are chosen to be those which provided the best
description of the HERA data on exclusive vector meson production in
Marquet:2007qa , $c=4.401$ and $B=3.713$, respectively.

Another parameter which enters the calculations is $R_{q}$, which relates the
dipole-proton and dipole-pion scattering amplitude, Eq.(7). Its value,
according to the additive quark model, is expected to be 2/3, the ratio
between the number of valence quarks in both targets. This was the same value
obtained for $R_{\pi/p}\equiv\sigma_{\pi p}/\sigma_{pp}$, the ratio between
the pion-proton and proton-proton cross sections, when the quark model was
applied to soft hadronic interactions Levin:1965mi . $R_{\pi/p}=2/3$ was also
experimentally observed in the low energy domain of hadronic reactions,
provided good description of previous ZEUS data on leading neutron spectra
Kaidalov2006 ; Khoze2006  and is supported by the investigation of the pion
structure function done in Nikolaev2000157 . On the other hand, when relation
(1) was applied to HERA data on photoproduction Chekanov2002 , the resulting
value for this ratio was $R_{\pi p}\simeq 1/3$ and, in recent work, it has
been concluded that it could reach 0.5 PhysRevD.85.114025 . Therefore, as we
can see, the value for the parameter $R_{q}$ is still an open question, and
what we can say is that it is expected to be in the range $1/2\leq R_{q}\leq
2/3$. Aiming at a comparison with the analysis done in coordinate space in
Goncalves:2015mbf , in the present analysis we will keep it fixed at
$R_{q}=2/3$.

The cross section for leading neutron production (1) depends on the pion flux,
whose generic form reads

$f_{\pi/p}=\frac{1}{4\pi}\frac{2g_{p\pi
n}^{2}}{4\pi}\frac{-t}{(t-m_{\pi}^{2})^{2}}(1-x_{L})^{1-2\alpha(t)}\left[F(x_{L},t)\right]^{2},$
(14)

where $g^{2}_{p\pi n}/4\pi=14.11$ is the $p\pi n$ coupling constant
Ericson:2000md , $\alpha(t)$ is the pion trajectory and $F(t,x_{L})$ is a
model-dependent form factor, which accounts for the finite size of the nucleon
and the pion. As in Goncalves:2015mbf , here we consider five parametrizations
for the form factor:

$F_{1}(x_{L},t)=\exp\left[R^{2}\frac{(t-m_{\pi}^{2})}{(1-x_{L})}\right],\quad\alpha(t)=0$
(15)

HOLTMANN1994363 , where $R=0.6$ GeV-1;

$F_{2}(x_{L},t)=1,\quad\alpha(t)=\alpha(t)_{\pi}$ (16)

BISHARI1972510 , where $\alpha(t)_{\pi}\simeq t$ ($t$ is given in GeV2);

$F_{3}(x_{L},t)=\exp\left[b(t-m_{\pi}^{2})\right],\quad\alpha(t)=\alpha(t)_{\pi}$
(17)

Kopeliovich1997 , where $\alpha(t)_{\pi}\simeq t$ ($t$ is given in GeV2) and
$b=0.3$ GeV-2;

$F_{4}(x_{L},t)=\frac{\Lambda_{m}^{2}-m_{\pi}^{2}}{\Lambda_{m}^{2}-t},\quad\alpha(t)=0$
(18)

PhysRevD.43.59 , where $\Lambda_{m}=0.74$ GeV;

$F_{5}(x_{L},t)=\left[\frac{\Lambda_{d}^{2}-m_{\pi}^{2}}{\Lambda_{d}^{2}-t}\right]^{2},\quad\alpha(t)=0$
(19)

PhysRevD.43.59 , where $\Lambda_{d}=1.2$ GeV. For each of these expressions
for the form factor one has a corresponding model for the pion flux, which we
will call, respectively, $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ and $f_{5}$.

Finally, a crucial point of the analysis of processes involving leading
neutron production is that the absorptive corrections must be taken into
account. These arise because the photon eventually also hits the neutron,
leading to extra interactions and, therefore to a reduction of the cross
section. Usually, these absorptive corrections are represented by a constant
factor $K$, which multiplies the (uncorrected) cross section (1). Following
the analysis done in Goncalves:2015mbf , we obtain the $K$-factor through the
ratio between the experimental and theoretical (calculated) total cross
sections. The uncertainty on the former translates into a range of possible
values for $K$, which will be around a central value $K_{\textrm{med}}$,
between a minimum value $K_{\textrm{min}}$ and a maximum value
$K_{\textrm{max}}$.

Table 1: $\chi^{2}/N_{\textrm{pts}}$ for all the different combinations of the models for the overlap functions and the pion flux $f_{\pi/p}$. | $\chi^{2}/N_{\textrm{pts}}$
---|---
Model for overlap functions |  $f_{1}$ | $f_{2}$ | $f_{3}$ | $f_{4}$ | $f_{5}$
BG | 1.178 | 0.724 | $0.559$ | 1.462 | 1.559
LCG | 1.170 | 0.718 | 0.559 | 1.468 | 1.564

Our first results are presented in Fig. 2, which shows the differential cross
section $\frac{d\sigma(W,Q^{2},x_{L})}{dx_{L}}$ as a function of $x_{L}$,
where the range in the leading neutron transverse momentum is $p_{T,n}<0.2$
GeV, one of the kinematical ranges covered by the data in Andreev2016 . We
consider the five models for the pion flux presented above and both BG and LCG
models for the overlap functions. The value of the resulting $K$-factor in
each case is also given. As in the coordinate-space analysis, for all the
models for the pion flux, the LCG model for the overlap functions tends to
provide larger values of $K$, in comparison with BG model. We can also observe
that, as in the coordinate-space case, model $f_{2}$ for the pion flux gives
the smallest value for $K$, while $f_{4}$ gives the largest one. Among the
five models for the pion flux, the one which provides the best description of
the data is $f_{1}$, as we can see in Table 1, where we present the calculated
$\chi^{2}$ over the number of points for all the combinations of models for
the pion flux and overlap functions. This is also in contrast with the
coordinate-space study, in which the best descriptions of the data were
obtained by using models $f_{2}$ and $f_{3}$. From Table 1 we can also see
that, in which concerns the goodness of the fit to the data, there is no
preference for on of the models for the overlap functions.

The most remarkable difference between our approach and the coordinate space
one is that the magnitude of the absorption corrections are quite different.
These corrections, mimicked by an overall constant $K$ factor, are related to
rescattering of the projectile photon. As in Goncalves:2015mbf , we estimated
its range of values by taking into account, as a constraint, the experimental
value for the total cross section of the process Andreev2016 . The results are
presented in Fig. 3, where we show our results using the model $f_{1}$ for the
pion flux, including the possible range of values for the $K$-factor, with
both BG and LCG models for the overlap functions. This range is
$K\mathrel{\hbox to0.0pt{\raise 2.20013pt\hbox{$>$}\hss}{\lower
2.20013pt\hbox{$\sim$}}}0.65$ to $K\mathrel{\hbox to0.0pt{\raise
2.20013pt\hbox{$<$}\hss}{\lower 2.20013pt\hbox{$\sim$}}}0.89$, when BG model
is used for the overlap functions, and $K\mathrel{\hbox to0.0pt{\raise
2.20013pt\hbox{$>$}\hss}{\lower 2.20013pt\hbox{$\sim$}}}0.70$ to
$K\mathrel{\hbox to0.0pt{\raise 2.20013pt\hbox{$<$}\hss}{\lower
2.20013pt\hbox{$\sim$}}}0.96$ when LCG model is used. This is in a strong
contrast to the results obtained in Goncalves:2015mbf , where it was found
that the $K$-factor falls in a quite different range, going from
$K\mathrel{\hbox to0.0pt{\raise 2.20013pt\hbox{$>$}\hss}{\lower
2.20013pt\hbox{$\sim$}}}0.11$ to $K\mathrel{\hbox to0.0pt{\raise
2.20013pt\hbox{$<$}\hss}{\lower 2.20013pt\hbox{$\sim$}}}0.16$ when using
$f_{2}$ and $K\mathrel{\hbox to0.0pt{\raise 2.20013pt\hbox{$>$}\hss}{\lower
2.20013pt\hbox{$\sim$}}}0.15$ to $K\mathrel{\hbox to0.0pt{\raise
2.20013pt\hbox{$<$}\hss}{\lower 2.20013pt\hbox{$\sim$}}}0.21$ when using
$f_{3}$ for the pion flux. Thus, in comparison with the coordinate study using
bCGC model, the present analysis, using the mixed-space MPS model, indicates
that absorption effects are substantially weaker and of the same order of the
corresponding corrections in semi-inclusive processes.

Figure 2: Leading neutron spectra in exclusive $\rho$ production considering
the boosted gaussian (left) and the light-cone Gauss (right) models for the
vector meson wave function for the five different models of the pion flux. The
data were taken from Andreev2016 .

We also compared our predictions to other H1 data in a different range of the
transverse momentum of the leading neutron, $p_{T,n}<0.69x_{L}$ GeV. Using the
model $f_{1}$ for the pion flux and following Goncalves:2015mbf , we fixed the
same ranges for $K$ obtained in Fig. 3 and use them to describe these data.
The results are shown in Fig. 4, for both BG and LCG models for the overlap
functions. The same procedure was done using all the other models for
$f_{\pi/p}$ and we obtained, again, that $f_{1}$ provides the best description
of the data, again with no preference between BG and LCG models for the
overlap functions.

Figure 3: Leading neutron spectra in exclusive $\rho$ production for the pion
flux model $f_{1}$ considering the boosted Gaussian (left) and the light-cone
Gauss (right) models for the vector meson wave function. The possible range of
values of the $K$ factor are shown. The data were taken from Andreev2016 .

Figure 4: Prediction for the leading neutron spectra in exclusive $\rho$
production considering $p_{T}<0.69\cdot x_{L}$ GeV. The possible range of
values of the $K$ factor for the pion flux model $f_{1}$ is fixed by the
previous set of experimental data. The data were taken from Andreev2016 .

##  V Conclusions

In this work we used a mixed-space approach of the color dipole formalism to
study the exclusive vector meson production in $ep$ collisions involving
leading neutron. For the mixed-space dipole-pion scattering amplitude we used
the MPS phenomenological model for the dipole-proton scattering amplitude,
which is based on the traveling wave solutions of BK equation, the simplest
nonlinear evolution equation of high energy QCD. Following the full
coordinate-space analysis done in Goncalves:2015mbf , we assumed a simple
relation between the dipole-pion and dipole-proton amplitudes, Eq.(7) with
fixed $R_{q}=2/3$, tested five models for the pion flux and confronted our
results to recent HERA data on exclusive $\rho$ meson production with leading
neutron. From the five models used for the pion flux, Eqs.(15)-(19), model
$f_{1}$ for the pion flux provided the best description of the data, in
contrast to the full coordinate-space analysis, in which the best descriptions
were obtained by using models $f_{2}$ and $f_{3}$ Eqs.(16) and (17).

We also estimated the magnitude of the absorption effects, usually included in
an overal $K$-factor, and our results indicate that in exclusive processes
involving leading neutrons the absorption effects are weaker in comparison
with the full coordinate-space approach. This can be verified by the possible
values for the $K$-factor, which according to our analysis should not be
smaller than $0.6$ (being of the order of those predicted for semi-inclusive
processes), while in the coordinate-space one they should not be larger than
$0.3$. It should be pointed out that here, as in Goncalves:2015mbf , the
values obtained for $K$ were directly related to the choice for the parameter
$R_{q}$, which in the coordinate approach translates into a large flexibility
in the values of the former as the latter goes from 1/3 to 2/3. This is not
the case in the present analysis, which clearly indicates that $R_{q}$ could
be smaller than, but not so far from, the value predicted by the additive
quark model.

From the above conclusions we can see that, as long as more data and
theoretical studies on exclusive processes with leading neutron become
available, we will be able to discriminate between both (coordinate and mixed
space) approaches and, therefore, shed new light on (or even reduce) other
uncertainties involving these processes, i.e., the more suitable model for the
pion flux, absorption corrections and the relation between the dipole-pion and
dipole-proton scattering amplitudes.

## ACKNOWLEDGMENTS

We would like to thank Victor Gonçalves and Diego Spiering for very useful
discussions. This work was partially supported by CAPES (Brazilian research
funding agency).

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