# Electromagnetic interaction of low transverse momentum electrons and
positrons
with heavy nuclei in ultra-peripheral ultra-relativistic heavy-ion collisions

K. Mazurek<EMAIL_ADDRESS>Institute of Nuclear Physics PAN, ul.
Radzikowskiego 152, Pl-31342 Kraków, Poland  M. Kłusek-Gawenda
<EMAIL_ADDRESS>Institute of Nuclear Physics PAN, ul.
Radzikowskiego 152, Pl-31342 Kraków, Poland  J. Józefiak<EMAIL_ADDRESS>AGH University of Science and Technology, al. Mickiewicza 30, Pl-30059 Kraków,
Poland  A. Szczurek<EMAIL_ADDRESS>Institute of Nuclear Physics
PAN, ul. Radzikowskiego 152, Pl-31342 Kraków, Poland  College of Natural
Sciences, Institute of Physics, University of Rzeszów, ul. Pigonia 1,
PL-35-310 Rzeszów, Poland

###### Abstract

The photon-photon interaction in ultra-relativistic heavy-ion collision is a
source of the $(e^{+},e^{-})$ pairs. The photon-photon fusion leads the lepton
creation in the broad configuration space around ”collision” point. Those
created close to heavy nuclei may undergo strong interaction with them. The
impact parameter space distribution of electrons and positrons are calculated
within the b-space EPA model of such collisions. The evolution due to the
electromagnetic interaction of $(e^{+},e^{-})$ with nuclei is studied, and the
distortion of rapidity and transverse momentum distributions are shown. We
show first exploratory results for the reaction Pb+Pb at $\sqrt{s_{NN}}=$ 17.3
GeV (SPS) and 200 GeV (RHIC) energies. We provide results for selected
creation points and when integrating over their position as dictated by the
$b$-space EPA model. We observe strong effects at low transverse momenta, so
far not measured regions of the phase space. In particular, we predict a
possible sizeable accumulation of electrons with rapidities close to the beam
rapidity. The EM effects lead to asymmetry in the production of electrons and
positrons.

##  I Introduction

Heavy-ion collisions are excellent factory for producing both elementary and
composed particles as well as for studying their properties and production
mechanism. Since many years efforts of theorists and experimentalists were
focused on the investigation of time-space evolution of the quark-gluon plasma
(QGP) and production of different species of particles, primarily hadrons
(pions, kaons, nucleons, hyperons, etc.) emitted in the collision. At high
energies, the velocities of such beam nuclei are close to light velocity thus
they are often called ultra-relativistic velocities (URV).

Central collisions are the most interesting in the context of the QGP studies.
Plasma is, of course, also produced in more peripheral collisions. In
peripheral collisions, the so-called spectators are relatively large and have
large moving charge. It was realized relatively late that this charge
generates strong quickly changing electromagnetic fields that can influence
the trajectories and some observables for charged particles.

Such effects were investigated in previous studies of one of the present
authors Rybicki:2006qm ; Rybicki:2013qla . On one hand side the EM effects
strongly modify the Feynman $x_{F}$ spectra of low-$p_{T}$ pions, creating a
dip for $\pi^{+}$ and an enhancement for $\pi^{-}$ at
$x_{F}\approx\frac{m_{\pi}}{m_{N}}$. In Rybicki:2006qm  a formalism of charged
meson evolution in the EM field (electric and magnetic) of fast moving nuclei
was developed. Later on the spectacular effects were confronted with the SPS
data Rybicki:2009zz  confirming the theoretical predictions. The investigation
was done for 208Pb+208Pb at 158 GeV/nucleon energy ($\sqrt{{s}_{NN}}=$ 17.3
GeV) at CERN Super Proton Synchrotron (SPS) Schlagheck:1999aq . In
Rybicki:2013qla  the influence of the EM fields on azimuthal flow parameters
($v_{n}$) was studied and confronted in Rybicki:2014rna  with the RHIC data.
It was found that the EM field leads to a split of the directed flow for
opposite e.g. charges. In the initial calculation, a simple model of single
initial creation point of pions was assumed for simplicity. More recently,
such a calculation was further developed by taking into account also the time-
space evolution of the fireball, treated as a set of firestreaks
Ozvenchuk:2019cve . The distortions of the $\pi^{+}$ and $\pi^{-}$
distributions allow to discuss the electromagnetic effects of the spectator
and charged pions in URV collisions of heavy ions and nicely explain the
experimental data. This study was done for non-central collisions where the
remaining after collision object, called ’spectator’, loses only a small part
of nucleons of the original beam/target nucleus.

The ultra-peripheral heavy-ion collisions (e.g. 208Pb+208Pb) at ultra-
relativistic energies ($\sqrt{s_{NN}}\geq 5~{}$GeV) Klusek-Gawenda:2016suk
allow to produce particles in a broad region of impact parameter space, even
far from “colliding” nuclei. The nuclei passing near one to each other with
ultrarelativistic energies are a source of virtual photons that can collide
producing e.g. a pair of leptons. In real current experiments (RHIC, LHC), the
luminosity is big enough to observe e.g. $AA\rightarrow AA\rho_{0}$ and
$AA\rightarrow AAe^{+}e^{-}$, $AA\rightarrow AA\mu^{+}\mu^{-}$ processes. One
of the most interesting phenomena is multiple interaction Klusek-
Gawenda:2016suk ; vanHameren:2017krz  which may lead to the production of more
than one lepton pair.

The studies on the creation of positron-electron pairs started in early
1930’ties with the discovery of positron by Dirac Dirac:1934  and works of G.
Breit and J.A. Wheeler Breit:1934zz , where they calculated the cross section
for the production of such pairs in the electric field. It was E.J. Williams
who realized Williams:1935  that the production of $e^{+}e^{-}$ pairs is
enhanced in the vicinity of the atomic nucleus. An overview of the theoretical
investigation of the $e^{+}e^{-}$ pairs creation in historical context was
presented by e.g. J.H. Hubbell Hubbel:2006  and the detailed discussion about
this process in physics and astrophysics was written by R. Ruffini et al.
Ruffini:2009hg .

The early analyses were done in the momentum space and therefore did not
include all details in the impact parameter space. An example of the
calculation where such details are taken into account can be found e.g. in
Klusek-Gawenda:2010vqb ; Klusek-Gawenda:2016suk . In ultra-peripheral
collision (UPC), the nuclei do not collide without loosing, in principle, any
nucleon. However, the electromagnetic interaction induced by fast moving
nuclei may cause excitation of the nuclei and subsequent emission of different
particles, in particular neutrons Klusek-Gawenda:2013ema  that can be measured
both at RHIC and at the LHC. Moreover the UPC are responsible not only for
Coulomb excitation of the spectator but also for the multiple scattering and
production of more than one dielectron pair vanHameren:2017krz . With large
transverse momentum cut typical at RHIC and the LHC the effect is not
dramatic.

Can the strong EM fields generated at high energies modify the
electron/positron distributions? No visible effect was observed for electrons
with $p_{T}>$ 1 GeV as discussed in vanHameren:2017krz , where the ALICE
distributions were confronted with the $b$-space equivalent photon
approximation (EPA) model. But the electromagnetic effect is expected rather
at low transverse momenta. According to our knowledge this topic was not
discussed in the literature. As the spectators, which in ultra-peripheral
collisions are almost identical to colliding nuclei, are charged, they can
interact electromagnetically with electrons and positrons as it was in the
case of pions. Similar effects as observed for pions may be expected also for
charged leptons. The motion of particles in EM field depends not only on their
charge but also on their mass. Thus the distortions of $e^{+}/e^{-}$
distributions should be different than those for $\pi^{+}/\pi^{-}$
distributions. Also the mechanism of production is completely different. In
contrast to pion production, where the emission site is well localized, the
electron-positron pairs produced by photon-photon fusion can be produced in a
broad configuration space around the “collision” point - point of the closest
approach of nuclei. A pedagogical illustration of the impact parameter
dependence can be found e.g. in Klusek-Gawenda:2010vqb .

From one side, the previous works Ozvenchuk:2019cve  and reference therein,
were connected with the electromagnetic effects caused by the emission of the
pions from the fireball region. From the other side, the model considered
Klusek-Gawenda:2010vqb ; Klusek-Gawenda:2018zfz ; Klusek-Gawenda:2020eja  can
correctly estimate the localization in the impact parameter space. The present
study will be focused on electromagnetic interaction between
electrons/positrons with highly positively charged nuclei.

Our approach consists of two steps. First, the $e^{+}e^{-}$ distributions are
calculated within EPA in terms of initial distributions in a given space point
and at a given initial rapidity and transverse momentum. Secondly, the space-
time evolution of leptons in the electromagnetic fields of fast moving nuclei
with URV is performed by solving relativistic equation of motion
Rybicki:2011zz .

In Section II the details of the calculation of the differential cross section
of $e^{+}e^{-}$ production will be presented. We will not discuss in detail
equation of motion which was presented e.g. in Rybicki:2011zz . The results of
the evolution of electrons/positrons in the EM field of “colliding” nuclei are
presented in Section III.

##  II Lepton pair production, equivalent photon approximation

The particles originated from photon-photon collisions can be created in full
space around excited nucleus thus first of all the geometry of the reaction
should be defined.

In the present study, the ultra-peripheral collisions (UPC) are investigated
in the reaction plane ($b_{x},b_{y}$) which are perpendicular to the beam axis
taken as $z$-direction.

The collision point ($b_{x}=0,b_{y}=0$) is time-independent center of mass
(CM) of the reaction as shown in Fig. 1. The impact parameter is fixed as
double radius of each (identical) Pb nucleus b=(13.95 fm, 14.05 fm). For
comparison we will show results also for b=(49.95 fm, 50.05 fm).

Four characteristic points ($\pm$15 fm, 0), (0, $\pm$15 fm), which are
discussed later are also marked in the figure. In the present paper we shall
show results for these initial emission points for illustrating the effect of
evolution of electrons/positrons in the EM field of nuclei.

Figure 1:  The impact parameter space and the five selected points
($b_{x},b_{y}$): (0, 0), ($\pm$15 fm,0), (0,$\pm$15 fm), for which the
distribution in rapidity and transverse momentum will be compared latter on in
the text, shown in the CM rest frame.

Usually the exclusive dilepton production was estimated by using the monopole
charge form factor which allows to reproduce correctly the total cross
section. The differential cross sections are more sensitive to details, thus
realistic charge form factor (Fourier transform of the charge distribution)
has to be employed. 111Double scattering production of positron-electron pairs
using the realistic charge form factor has been discussed in Klusek-
Gawenda:2016suk  and vanHameren:2017krz . The total cross section for the
considered process ($AA\to AAe^{+}e^{-}$) can be written as:

$\displaystyle\sigma_{A_{1}A_{2}\rightarrow
A_{1}A_{2}e^{+}e^{-}}(\sqrt{s_{A_{1}A_{2}}})=$ (1) $\displaystyle=$
$\displaystyle\int\frac{d\sigma_{\gamma\gamma\rightarrow
e^{+}e^{-}}(W_{\gamma\gamma})}{d\cos{\theta}}N(\omega_{1},b_{1})N(\omega_{2},b_{2})S^{2}_{abs}(b)$
$\displaystyle\times$ $\displaystyle 2\pi
bdbd\overline{b_{x}}d\overline{b_{y}}\frac{W_{\gamma\gamma}}{2}dW_{\gamma\gamma}dY_{e^{+}e^{-}}\
d\cos{\theta},$

where $N(\omega_{i},b_{i})$ are photon fluxes,
$W_{\gamma\gamma}=M_{e^{+}e^{-}}$ is invariant mass and
$Y_{e^{+}e^{-}}=(y_{e^{+}}+y_{e^{-}})/2$ is rapidity of the outgoing system
and $\theta$ is the scattering angle in the $\gamma\gamma\rightarrow
e^{+}e^{-}$ center-of mass system. The gap survival factor $S^{2}_{abs}$
assures that only ultra-peripheral reactions are considered.

Figure 2: The map of differential cross section in rapidity of electron or
positron and lepton transverse momentum for ($b_{x},b_{y}$) = (0, 0) which
will be called CM point for brevity.

The $\overline{b_{x}}=(b_{1x}+b_{2x})/2$ and
$\overline{b_{y}}=(b_{1y}+b_{2y})/2$ quantities are particularly useful for
our purposes. We define $\vec{{\bar{b}}}=(\vec{b}_{1}+\vec{b}_{2})/2$ which is
(initial) position of electron/positron in the impact parameter space. This
will be useful when considering motion of electron/positron in the EM field of
nuclei. The energies of photons are included by the relation:
$\omega_{1,2}=W_{\gamma\gamma}/2\exp(\pm Y_{e^{+}e^{-}})$. In the following
for brevity we shall use $b_{x},b_{y}$ instead of
$\overline{b_{x}},\overline{b_{y}}$. Then $(b_{x},b_{y})$ is the position in
the impact parameter plane, where the electron and positron are
created.222Expression (1) allows to estimate not only the lepton pair
production but also a production of any other particle pair Klusek-
Gawenda:2010vqb . The differential (in rapidity and transverse momentum) cross
section could be obtained in each emission point in the impact parameter space
($b_{x},b_{y}$).

Figure 3: (Color on-line) The differential cross section for various emission
points of electrons/positrons produced in the 208Pb+208Pb reaction at 158
GeV/nucleon energy ($\sqrt{{s}_{NN}}=$ 17.3 GeV) at impact parameter
14$\pm$0.05 fm. The cross section for selected points ($b_{x},b_{y}$): (0, 0),
($\pm$15 fm, 0), (0, $\pm$15 fm) and (40 fm, 0) are integrated over $p_{T}$
(a) and rapidity (b), respectively.

The calculations will be done assuming the collision of 208Pb+208Pb at 158
GeV/nucleon energy ($\sqrt{{s}_{NN}}=$ 17.3 GeV) corresponding to the CERN SPS
and $\sqrt{{s}_{NN}}=$ 200 GeV of the STAR RHIC at impact parameter
14$\pm$0.05 fm which is approximately twice the radius of the lead nucleus.
This is minimal configuration assuring ultra-peripheral collisions.

Figure 2 illustrates the differential cross section on the plane of rapidity
($y$) vs. transverse momentum ($p_{T}$). Rather broad range of rapidity (-5,
5) is chosen, but the distribution in $p_{T}$ will be limited to (0, 0.1 GeV)
as the cross section drops at $p_{T}$ = 0.1 GeV already a few orders of
magnitude. The electromagnetic effects may be substantial only in the region
of the small transverse momenta.

Thus for our exploratory study here we have limited the range for rapidity to
(-5, 5) and for transverse momentum to $p_{T}$=(0, 0.1 GeV). The integrated
distribution can be seen in Fig. 3(a) and (b). There we compare the
distributions obtained for different emission points ($b_{x},b_{y}$): (0, 0),
($\pm$15 fm, 0), (0, $\pm$15 fm) as shown in Fig. 1. The behavior of the
differential cross section is very similar in each ($b_{x},b_{y}$) point but
it differs in normalization as it is shown in Fig. 3.

Figure 4:  Distribution of the cross section in the impact parameter $b$ for
different energies: $\sqrt{s_{NN}}$ = 17.3, 50 and 200 GeV (from bottom to
top).

In Fig. 4 we show a distribution of the cross section in impact parameter $b$
for different collision energies $\sqrt{s_{NN}}$ = 17.3, 50, 200 GeV. In this
calculation we have taken $p_{T}>$ 0 GeV (the cross section strongly depends
on the lowest value of lepton transverse momentum $p_{T}$). In general, the
larger collision energy the broader the range of impact parameter. However,
the cross section for $b\approx R_{A_{1}}+R_{A_{2}}$ is almost the same. Only
taking into account limitation, e.g. on the momentum transfer, makes the
difference in the cross section significant even at $b=14$ fm.

(a)

(b)

Figure 5:  Two-dimensional cross section as a function of $b_{x}$ and $b_{y}$
for two values of impact parameter: (a) b=14$\pm$0.05 fm and (b) b=50$\pm$0.05
fm.

The emission point of the electrons/positrons does not change the behavior
(shape) of the cross section on the ($y,p_{T}$) plane but changes the absolute
value of the cross section. As it is visible in Fig. 3 (a) and (b) the biggest
cross section is obtained for the CM emission point. The production of
$e^{+},e^{-}$ at ($b_{x},b_{y}$) = (40 fm, 0) i.e. far from the CM point, is
hindered by three orders of magnitude. Moreover, the production at
$b_{x}$=$\pm$15 fm and $b_{y}$=0 is more preferable than the production at
$b_{x}$=0 and $b_{y}$=$\pm$15 fm what is fully understandable taking into
account the geometry of the system (see Fig. 1). As the system taken here into
consideration is fully symmetric ($A_{1}=A_{2},Z_{1}=Z_{2}$), thus
corresponding results are symmetric under the following replacements:
$b_{x}\to-b_{x}$ or $b_{y}\to-b_{y}$.

Figure 5 compares the integrated cross section on reaction plane
($b_{x},b_{y}$) for two impact parameters: (a) b=14$\pm$0.05 fm (when nuclei
are close to each other) and (b) b=50$\pm$0.05 fm (when nuclei are well
separated). The landscape reflects the position of the nuclei in the moment of
the closest approach. Similar plots have also been done for higher
$\sqrt{s_{NN}}$ but the shape is almost unchanged, only the cross section
value is different. This figure illustrates the influence of the geometry of
the reaction. Regardless of the impact parameter (b), distance between
colliding nuclei, the cross-section has a maximum at $b_{x}=0$. The change of
$b$ is correlated with the shift in a peak at $b_{x}$.

Figure 6: (Color on-line) The differential cross section for various emission
points of electrons in the 208Pb+208Pb reaction at $\sqrt{{s}_{NN}}=$ 17.3,
50, 200 GeV at impact parameter 14$\pm$0.05 fm.

The calculations confirm that the shape of the electron/positron distribution,
shown in Fig. 6 does not depend on the energy of the colliding nuclei. There
are visible small differences in the magnitude of cross sections for
$\sqrt{{s}_{NN}}=$ 17.3 and 200 GeV at least in the selected limited
$p_{T}$=(0, 0.1) GeV range. Dependence on the rapidity is even weaker as the
differences are visible only for $|y|>$3.

These cross sections are used as weights in calculation of electromagnetic
effects between electrons/positrons and the fast moving nuclei. The
corresponding matrix has following dimensions: $b_{x,y}$=(-50 fm, 50 fm) -
99$\times$99 points in the reaction plane and 100$\times$15 in the ($y,p_{T}$)
space.

Figure 7:  Distribution of the cross section in $\log_{10}p_{T}$ for different
energies: $\sqrt{s_{NN}}$ = 17.3, 50 and 200 GeV (from bottom to top).

Rather small transverse momenta enter such calculation. To illustrate this in
Fig. 7 we show a distribution in $log_{10}(p_{T})$. As seen from the figure
the cross section is integrable and we have no problem with this with our
Monte Carlo routine Lepage:1977sw .

##  III Electromagnetic interaction effects

The spectator system are modeled as two uniform spheres in their respective
rest frames that change into disks in the overall center-of-mass collision
frame. The total charge of nuclei is 82 consistent with UPC. The lepton
emission region is reduced to a single point and the time of emission is a
free parameter.

(a) ($b_{x}$=0, $b_{y}$=0) (b) ($b_{x}$=0, $b_{y}$=15 fm) (c) ($b_{x}$=15 fm,
$b_{y}$=0)

(d) full ($b_{x}$, $b_{y}$) space (e) ($b_{x}$=0, $b_{y}$=-15 fm) (f)
($b_{x}$=-15 fm, $b_{y}$=0)

(a) ($b_{x}$=0, $b_{y}$=0) (b) ($b_{x}$=0,$b_{y}$=15 fm) (c) ($b_{x}$=15 fm,
$b_{y}$=0)

(d) full ($b_{x}$, $b_{y}$) space (e) ($b_{x}$=0, $b_{y}$=-15 fm) (f)
($b_{x}$=-15 fm, $b_{y}$=0)

Figure 8: (Color on-line) Rapidity vs $p_{T}$ distributions for final
(subjected to EM effects) electrons for different emission points: (a) (0, 0)
and (d) full xy plane; (b) (0, 15 fm); (c) (15 fm, 0); (e) (0, -15 fm) and (f)
(-15 fm, 0). These results are for $\sqrt{s_{NN}}$ = 17.3 GeV. Figure 9:
(Color on-line) Rapidity vs $p_{T}$ distributions for final (subjected to EM
effects) positrons for different emission points: (a) (0, 0) and (d) full xy
plane; (b) (0, 15 fm); (c) (15 fm, 0); (e) (0, -15 fm) and (f) (-15 fm, 0).
These results are for $\sqrt{s_{NN}}$ = 17.3 GeV.

(a) $\sqrt{s_{NN}}$= 17.3 GeV (b) $\sqrt{s_{NN}}$= 17.3 GeV
b=14 fm ($b_{x}$=0,$b_{y}$=0) b=50 fm

(c) $\sqrt{s_{NN}}$= 50 GeV (d) $\sqrt{s_{NN}}$= 200 GeV
b=14 fm b=14 fm

Figure 10: (Color on-line) Reduced rapidity distributions for final electrons
(blue) and positrons (red) for fixed $b$ and ($b_{x}$=0,$b_{y}$=0) plane of
emission points at three collision energies: $\sqrt{s_{NN}}$=17.3, 50 and 200
GeV.

In this work we assume there is no delay time between collisions of nuclei and
the start of the EM interactions. The $z$-dependence of the first occurrence
of the $e^{+}e^{-}$ pair is beyond the EPA and is currently not known. In our
opinion production of $e^{+}e^{-}$ happens when the moving cones, fronts of
the EM fields, cross each other. This happens for $z\approx$ 0\. In the
following we assume $z$ = 0 for simplicity. 333Any other distribution could be
taken.

(a) $\sqrt{s_{NN}}$= 17.3 GeV (b) $\sqrt{s_{NN}}$= 17.3 GeV
full ($b_{x}$,$b_{y}$) space
b=14 fm b=50 fm

(c) $\sqrt{s_{NN}}$= 50 GeV (d) $\sqrt{s_{NN}}$= 200 GeV
b=14 fm b=14 fm

Figure 11: (Color on-line) Reduced rapidity distributions for final electrons
(blue) and positrons (red) for fixed $b$ and full ($b_{x}$,$b_{y}$) plane of
emission points at three collision energies: $\sqrt{s_{NN}}$=17.3, 50 and 200
GeV.

The trajectories of $e^{\pm}$ in the field of moving nuclei are obtained by
solving the equation of motion numerically for electrons/positrons:

$\frac{d\vec{p}_{e^{\pm}}}{dt}=\vec{F}_{1,e^{\pm}}(\vec{r}_{1},t)+\vec{F}_{2,e^{\pm}}(\vec{r}_{2},t)\;.$
(2)

The total interaction is a superposition of interactions with both nuclei
which positions depend on time. We solve the motion of electron/positron in
the overall center of mass system, i.e. both position and time are given in
this frame. In this frame we have to deal with both electric and magnetic
force Rybicki:2006qm . Because nuclei are very heavy compared to
electrons/positron their motion is completely independent and is practically
not distorted by the EM interaction. We take:

$\displaystyle\vec{r}_{1}(t)=+{\hat{z}}ct+\vec{b}/2\;,$
$\displaystyle\vec{r}_{2}(t)=-{\hat{z}}ct-\vec{b}/2\;,$ (3)

i.e. assume that the nuclei move along straight trajectories independent of
the motion of electron/positron. The step of integration depends on energy and
must be carefully adjusted.

The rapidity vs $p_{T}$ distributions of initial leptons are obtained by
randomly choosing the position on the two-dimensional space. The path of
particle in electromagnetic field generated by nuclei are traced up to 10 000
fm away from the original interaction point. The Monte Carlo method is used to
randomize the initial rapidity and $p_{T}$ from uniform distribution. The
initial rapidity and $p_{T}$ of electrons/positrons are randomly chosen in the
range: $y$=(-5,5) and $p_{T}$=(0, 0.1) GeV as fixed in the previous section.
The ($y,p_{T}$) distributions for final (subjected to the EM evolution)
leptons are presented in Fig. 9 for electrons and in Fig. 9 for positrons.
These distributions were obtained by analyzing the EM evolution event-by-
event. The number of events taken here is $n_{event}$ = 107 for each two-
dimensional plot.

For electron production an enhancement and for positron a loss with respect to
the neighborhood or/and flat initial distribution is observed for
$y\approx\pm$ 3\. This corresponds to the beam rapidity at $\sqrt{s_{NN}}$=
17.3 GeV energy.

These two sets of two-dimensional plots illustrate effect of the EM
interaction between $e^{+}$ or $e^{-}$ and the moving nuclei. The motion of
particles in the EM field of nuclei changes the initial conditions and the
final ($y,p_{T}$) are slightly different. Fig. 9 shows the behavior of
electrons and Fig. 9 of positrons at CM (a) and different impact parameter
points: (panels c,f) ($\pm$15 fm,0) and (panels b,e) (0,$\pm$15 fm) marked in
Fig. 1. We observe that the maximal number of electrons is located where the
cross section for positrons has minimum. The Coulomb effects are well visible
as a missing areas for positrons for $p_{T}<0.02$ GeV for particles emitted
from the CM point. The emission of leptons from $b_{y}$=$\pm$15 fm gives lower
effect. The asymmetry in emission is well visible for $b_{x}$=$\pm$15 fm,
where a larger empty space is for positive rapidity when $b_{x}$=15 fm and for
negative rapidity when $b_{x}$=-15 fm.

Although the EM effects are noticeable for electrons/positrons in different
impact parameter points, the integration over full reaction plane washes out
almost totally the effect.

For comparison the results of integration over full space $b_{x}$=(-50 fm, 50
fm) and $b_{y}$=(-50 fm, 50 fm) are shown in panels (d) of Figs. 9 and 9.
These results are independent of the source of leptons, thus it could be
treated as a general trend and an indication for which rapidity-transverse
momentum ranges one could observe effects of the EM interaction between
leptons and nuclei.

(a) $\sqrt{s_{NN}}$= 17.3 GeV (b) $\sqrt{s_{NN}}$= 17.3GeV
b=14 fm b=50 fm

(c) $\sqrt{s_{NN}}$= 50 GeV (d) $\sqrt{s_{NN}}$= 200 GeV
b=14 fm b=14 fm

Figure 12: (Color on-line) Transverse momentum distributions for final
electrons (blue) and positrons (red) for fixed $b$ and full ($b_{x}$, $b_{y}$)
plane of emission points at three collision energies: $\sqrt{s_{NN}}$=17.3, 50
and 200 GeV.

The distribution in reduced rapidity of final leptons are shown in Fig. 10 and
11. The reduced rapidity (dimensionless quantity) is the rapidity $y$
normalized to the beam rapidity $y_{beam}$ (different for various collision
energies)

$y_{red}=y/y_{beam}\;,$ (4)

where

$y_{beam}=\pm ln\left(\frac{\sqrt{s_{NN}}}{m_{p}}\right)$ (5)

and $m_{p}$ is the proton mass. The results shown above were obtained,
somewhat arbitrarily, with uniform distribution in $(y,p_{T})$. This leads to
the observation of peaks or dips at beam rapidities. No such peaks appear for
$\sqrt{s}$ = 200 GeV as here the chosen range of rapidity (-5,5) is not
sufficient. Whether such effects survive when weighting with the b-space EPA
cross section will be discussed below.

Fig. 10 is focused on emission from the center of mass point and Fig. 11 is
obtained when integrating over full ($b_{x},b_{y}$) plane. The main
differences between electron (blue lines) and positron (red lines)
distributions are not only at midrapidities but also around the beam rapidity.
The effect is more visible for the CM emission point but it is slightly
smoothed out when the full $(b_{x},b_{y})$ plane is taken into consideration.
Moreover increasing the impact parameter (panels (a) and (b)) diminishes the
difference between rapidity distributions of final electrons and positrons.
The beam energy is another crucial parameter. The collision with
$\sqrt{s_{NN}}>100$ GeV (panel (d)) does not allow for sizeable effects of
electromagnetic interaction between leptons and nuclei, at least at
midrapidities.

The discussion of the EM interaction between $e^{+}e^{-}$ and nuclei has to be
completed by combining with the cross section of lepton production as obtained
within EPA. Taking into account the leptons coming from photon-photon fusion
the distributions from Fig. 9 and 9 are multiplied by differential cross
section obtained with Eq.( 1).

The details of the method are presented in Ref. Rybicki:2006qm  and adapted
here from pion emission to electron/positron emission.

Figure 13: (Color on-line) The electron and positron emission cross section
normalized to 100% in the 208Pb+208Pb reaction at 158 GeV/nucleon energy
($\sqrt{{s}_{NN}}=$ 17.3 GeV) at impact parameter 14$\pm$0.05 fm assuming the
$p_{T}$=(0, 0.1) GeV produced in the center of mass (0, 0) point (a) and when
integrating over full reaction space (b). Shown are original EPA distributions
(dotted line) and results when including evolution in the EM field of nuclei
for positrons (solid line) and for electrons (dashed line).

In Fig. 12 we show the influence of EM interaction on $p_{T}$ distributions.
Here we integrate over rapidity and ($b_{x},b_{y}$). One can observe that the
EM effects lead to a diffusion of transverse momenta (see the diffused edge at
$p_{T}=0.1$ GeV, marked by green vertical line). No spectacular effect is
observed when changing the impact parameter or beam energy.

Fig. 13 shows a comparison of rapidity distribution of final electrons and
positrons, assuming the particles are emitted from (a) the center of mass (0,
0) point and (b) when integrating over $b_{x},b_{y}$. The comparison is done
between EPA distribution relevant for initial stage (red, dotted line) with
the final stage, resulting from the EM interaction of charged leptons with
positively charged nuclei. If leptons are produced in the CM point, the
electron distributions are almost unchanged but positron distributions are
squeezed to $|y|<2$. If the cross section is integrated over full
($b_{x},b_{y}$) parameter space, the positron distribution is still steeper
than that for electrons but mainly for $|y|<$ 2.

Even when the leptons produced in the full ($b_{x},b_{y}$) plane are
considered, the $e^{+}$ and $e^{-}$ distributions are different from the
initial ones. The electrons under the EM interactions are focused at
midrapidities.

(a) (b)

(c) (d)

Figure 14:  Distribution of electrons ((a), (c)) and positrons ((b), (d)) for
$\sqrt{s_{NN}}$= 17.3 GeV at b=14 fm integrated over ($b_{x},b_{y}$)=(-50 fm,
50 fm), $p_{T}^{ini}$=(0, 0.1 GeV)

(a) (b)

(c) (d)

Figure 15:  Distribution of electrons ((a), (c)) and positrons ((b), (d)) for
$\sqrt{s_{NN}}$= 17.3 GeV at b=50 fm integrated over ($b_{x},b_{y}$)=(-100 fm,
100 fm), $p_{T}^{ini}$=(0, 0.1 GeV).

(a) (b)

(c) (d)

Figure 16:  Distribution of electrons ((a), (c)) and positrons ((b), (d)) for
$\sqrt{s_{NN}}$= 200 GeV at b=14 fm integrated over ($b_{x},b_{y}$)=(-50 fm,
50 fm) for $p_{T}^{ini}$=(0, 0.1 GeV)

The dependence on the position of emission and final rapidity allows to
understand how the geometry influences the electromagnetic interaction between
leptons and nuclei. Figures 14,15 and 16 present the cross section
distribution in $b_{x}$ (top rows) and rapidity for electrons (a) and
positrons (b) and (bottom rows) $b_{y}$ and rapidity for electrons (c) and
positrons (d). Figs. 14 and 15 are for $\sqrt{s_{NN}}$= 17.3 GeV but with the
impact parameter 14$\pm$0.05 and 50$\pm$0.05 fm. Fig. 16 is for
$\sqrt{s_{NN}}$= 200 GeV and b=14 fm. These plots allow to investigate the
anisotropy caused by the interaction between leptons and nuclei. It is more
visible for larger impact parameter when the spectators are well separated
(Fig. 15).

Figure 17: Rapidity distribution of electrons for $\sqrt{s_{NN}}$= 17.3 GeV
(b=14 fm, 50 fm) and 50 GeV and 200 GeV with b=14$\pm$0.05 fm (only)
integrated over ($b_{x},b_{y}$)=(-50 fm, 50 fm), $p_{T}^{ini}$=(0, 0.1 GeV).
Figure 18: The ratio of rapidity distributions of positrons and electrons for
$\sqrt{s_{NN}}$= 17.3 GeV and fixed b=14 fm and 50 fm and $\sqrt{s_{NN}}$ = 50
GeV and 200 GeV with fixed b=14 fm integrated over ($b_{x},b_{y}$)=(-50 fm, 50
fm) and transverse momenta in the interval $p_{T}^{ini}$=(0, 0.1 GeV).

Distributions for fixed impact parameter and different beam energies reflect
the behavior seen in Fig. 11. For collision with $\sqrt{s_{NN}}$= 200 GeV
(Fig. 16) the electrons and positrons almost do not feel the presence of the
EM fields of the nuclei.

Integrating over full ($b_{x},b_{y}$) plane one obtains the rapidity
distribution of final leptons shown in Fig. 17 (a) separately for electrons
and (b) positrons, for two impact parameters: 14 fm (full lines) and 50 fm
(dashed lines). Electrons have a somewhat wider distribution than positrons
and this is independent of the impact parameter. While positron rapidity
distributions only weakly depend on collision energy it is not the case for
electrons, where sizeable differences can be observed.

The ratio of distributions for positrons and electrons (Fig. 18) reflects the
behavior seen in Figs. 14 and 15. This plot shows a combined effect of EPA
cross section and the EM interactions of leptons with nuclei. Thus despite the
production cross section is lower for larger impact parameter, the discussed
phenomenon should be visible for larger rapidities. The ratio quickly changes
with energy and tends to 1 for larger energies (see dotted line for
$\sqrt{{s}_{NN}}$ = 200 GeV).

##  IV Conclusions

In the last 15 years the electromagnetic effects due to a large (moving)
charge of the spectators on charged pion momentum distributions were observed
both in theoretical calculations and experimentally in peripheral heavy ion
collisions at SPS and RHIC energies. Interesting and sometimes spectacular
effects were identified.

In the present paper we have discussed whether such effects could also be
observed for the distributions of electrons/positrons produced via photon-
photon fusion in heavy ion UPC. The corresponding cross section can be rather
reliably calculated and turned out to be large, especially for low transverse
momentum electrons/positrons. The impact parameter equivalent photon
approximation is well suitable for investigating the electromagnetic effects.
On the experimental side only rather large transverse momentum
electrons/positrons could be measured so far at RHIC and the LHC, typically
larger than 0.5 GeV.

We have organized calculations that include the EM effects using as an input
EPA distributions. First multidifferential (in momenta and impact parameter)
distributions for the diphoton production of the $e^{+}e^{-}$ pairs are
prepared in the impact parameter equivalent photon approximation. Such
distributions are used next to calculate the propagation of the
electrons/positrons in strong EM fields generated by the quickly moving
nuclei. The propagation has been done by solving numerically relativistic
equation of motion. Strong EM effects have been observed only at very small
transverse momenta of electron/positron. Therefore to accelerate calculations
we have limited to really small initial transverse momenta $p_{T}<$ 0.1 GeV.

The shape of the differential cross section in rapidity and transverse
momentum does not depend on the energy of the process but rather on the
emission point in the impact parameter plane ($b_{x},b_{y}$). We have
investigated effects for different initial conditions, i.e. different emission
positions in the impact parameter space.

The leptons interact electromagnetically with charged nuclei which changes
their trajectories. The biggest effect has been identified for the CM emission
point. However, the integration over full ($b_{x},b_{y}$) plane washes out
this effect to large extent. The range of $p_{T}$=(0, 0.02 GeV) has turned out
the most preferable to investigate the influence of the EM effects on leptons
originating from various ($b_{x},b_{y}$) plane points.

Moreover, the impact parameter influences not only the value of the cross
section but also the shapes of distributions of final leptons.

The $AA\rightarrow AAe^{+}e^{-}$ process creates leptons in a broad range of
rapidities. We have found that only at small transverse momenta of
electrons/positrons one can observe sizeable EM effects.

The performed calculations allow to conclude that the maximal beam energy for
the Pb+Pb collision, where the EM effects between leptons and nuclei are
evident at midrapidities, is probably $\sqrt{{s}_{NN}}$=100 GeV. Observation
of the effect at higher energies may be therefore rather difficult, if not
impossible. The effect survives even at high energies but close to beam
rapidity. However, this region of the phase space is usually not instrumented
and does not allow electron/positron measurement.

So far the effect of EM interaction was studied for fixed values of the impact
parameter (mainly for $b$ = 14 fm). However, the impact parameter cannot be
measured. The integration over the impact parameter is rather difficult and
goes beyond the scope of the present paper. Such an integration will be
studied elsewhere.

On the experimental side a good measurement of electrons and positrons at low
transverse momenta ($p_{T}<$ 0.1 GeV is necessary to see an effect. In
principle, the measurement of the $e^{+}/e^{-}$ ratio as a function of lepton
rapidity and transverse momentum would be useful in this context. According to
our knowledge there are definite plans only for high energies (ALICE-3
project) where, however, the EM effect should be very small (high collision
energy). At CERN SPS energies the effect is rather large but at very small
transverse momenta. RHIC would probably be a good place to observe the effect
of the discussed here EM interactions but this would require a modification of
the present apparatus.

Acknowledgement

A.S. is indebted to Andrzej Rybicki for past collaboration on electromagnetic
effects in heavy ion collisions. This work is partially supported by the
Polish National Science Centre under Grant No. 2018/31/B/ST2/03537 and by the
Center for Innovation and Transfer of Natural Sciences and Engineering
Knowledge in Rzeszów (Poland).

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