# Electromagnetic fields in low-energy heavy-ion collisions with baryon
stopping

Ankit Kumar Panda<EMAIL_ADDRESS>Partha Bagchi
<EMAIL_ADDRESS>Hiranmaya Mishra<EMAIL_ADDRESS>Victor Roy
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science
Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India

(August 28, 2024)

###### Abstract

We investigate the impact of baryon stopping on the temporal evolution of
electromagnetic fields in vacuum at low-energy Au+Au collisions with
$\sqrt{s_{NN}}=4$-$20$ GeV. Baryon stopping is incorporated into the Monte-
Carlo Glauber model by employing a parameterized velocity profile of
participant nucleons with non-zero deceleration. The presence of these
decelerating participants leads to noticeable changes in the centrality and
$\sqrt{s_{NN}}$ dependence of electromagnetic fields compared to scenarios
with vanishing deceleration. The influence of baryon stopping differs for
electric and magnetic fields, also exhibiting variations across their
components. We observe slight alteration in the approximate linear dependency
of field strengths with $\sqrt{s_{NN}}$ in the presence of deceleration.
Additionally, the longitudinal component of the electric field at late times
becomes significant in the presence of baryon stopping.

††preprint: APS/123-QED

##  I Introduction

In an off-central relativistic heavy ion collision, intense transient
electromagnetic fields are produced predominantly due to the motion of
spectators, reaching magnitudes of approximately $\sim 10m_{\pi}^{2}$ at top
RHIC energies [1, 2, 3, 4, 5, 6, 7, 8]. Notably, the peak magnitude of event-
averaged values of the electromagnetic field produced in these collisions
shows an approximate proportionality to the square root of the center-of-mass
energy per nucleon pair, $\sqrt{s_{NN}}$ [4]. These results, however, were
obtained on the assumption of near-perfect transparency of the colliding
nucleons, as described by the Glauber model [9]. The Glauber model of nucleus-
nucleus collisions gives the number of binary collisions and spectators for a
given inelastic nucleon-nucleon cross section that depends on $\sqrt{s_{NN}}$
but is assumed to be independent of the number of binary collisions. For high
$\sqrt{s_{NN}}$, such as top RHIC and LHC energies, the elastic or diffractive
dissociation collisions lead to a minute loss of energy of the colliding
nucleon, and hence it is a good approximation to consider that the colliding
nucleons move in a straight line with almost constant velocity even after
multiple collisions. However, for low $\sqrt{s_{NN}}$ collisions, baryon
stopping can be sizeable [10, 11, 12, 13]. The baryon stopping must be taken
into account while estimating the electromagnetic fields using the Glauber
model at low $\sqrt{s_{NN}}$. Particularly, the temporal evolution of the
fields post-collisions will be affected due to the decelerations of the
protons after each binary collision. Previous studies show that, on average, a
proton loses half of its pre-collision energy, which is about one unit of
rapidity in each binary collision [14]. The maximum value of the net baryon
density at midrapidity is achieved for $\sqrt{s_{NN}}\approx 6$ GeV [15, 16].
Above this collision energy, the midrapidity net baryon density decreases with
increasing energy due to the higher transparency of the colliding nuclei.
Hence, for higher collision energies, the effect of deceleration in
calculating electromagnetic fields is minimal. The decelerated motion of
charged protons may additionally give rise to bremsstrahlung radiations that
could contribute to direct photon production along with other known
mechanisms. Experimental measurements on baryon stopping gives an exponential
distribution of net baryon density: A exp($-\alpha_{B}$$\delta$y) where
$\delta$y = $Y_{\text{beam}}$\- $Y_{\text{CM}}$ is the rapidity loss and the
exponent $\alpha_{B}$ ranges between 0.65-0.67 [11, 17]. Some theoretical
model studies including models based on Color Glass Condensate, could
successfully describe these data [18, 19, 20]. However, in the current study,
we do not use any of these models; instead, we take a more pedagogical
approach and parameterize the decelerated motion of participants in the Monte-
Carlo Glauber (MCG) model to mimic baryon stopping.

The calculation of precise space-time evolution of electromagnetic fields is
vital for studying anomaly-induced parity-violating phenomena in QCD, such as
the chiral magnetic effect (CME), where a charge/chiral current current
$\bm{j}/\bm{j_{5}}=\frac{e^{2}}{2\pi^{2}}(\mu_{5}/\mu)\bm{B}$ (here $\mu_{5}$
and $\mu$ are chiral and charge chemical potential respectively) is induced
along magnetic fields in a system with chiral fermions. This phenomenon has
attracted significant attention since its inception [21, 22, 23, 24, 25, 26,
27, 28]. As an experimental signature the charge-dependent two- and three-
particle correlations in Xe-Xe collisions has been proposed as a signature to
the presence of CME. However, one needs to separate the signal of CME from the
possible background and compare the experimental data with theoretical model
studies that take into account the correct spatiotemporal evolution of
electromagnetic fields. The lifetime and intensity of the electromagnetic
fields in heavy-ion collisions and hence the CME signal depends on the center
of mass energy of the collision and the chiral chemical potential of the hot
and dense nuclear matter, among other deciding factors. The Beam Energy Scan
(BES) program at RHIC and the future Facility for Antiproton and Ion Research
(FAIR) at GSI aim to explore the properties of the QCD matter at several low
center of mass energy collisions. These experiments will also provide a unique
opportunity to study whether the CME and related phenomena could still be
observed in lower-energy heavy-ion collisions.

As mentioned earlier, we need a dynamical model that captures the space-time
evolution of electromagnetic fields with the initial fields calculated from an
initial model such as MC Galuber and solve the Maxwell and fluid equations
self-consistently to represent the space-time evolution faithfully. As a first
step towards that direction, we study the temporal evolution of
electromagnetic fields in vacuum for lower $\sqrt{s_{NN}}$ collisions with
baryon stopping. We mimic the effect of baryon stopping via a parameterized
velocity profile (with non-zero deceleration) of nucleons undergoing binary
collisions.

Earlier studies related to the evolution of electromagnetic field in
conducting medium [5, 29] at rest and in an expanding charged conducting fluid
[30] show that the fields are sustained for a more extended period of time
compared to the vacuum evolution. Further, we expect a longer-lasting
electromagnetic field in low-energy collisions, albeit with a smaller peak
value, which is an interesting unexplored territory that we wish to explore.
Specifically, the effect of fields on anisotropic flow in low-energy heavy-ion
collisions is worthwhile to study as their high-energy counterpart shows some
interesting dependencies on flow coefficients and transport properties [31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. These
studies will give us unique access to measure the transport coefficients of
hot hadronic gas.

This paper is organized as follows: In the first section, in Sec.(II), we
discuss the theoretical formulation and the assumptions made for introducing
the stopping of baryons after the collisions. Then, in Sec.(III), we briefly
discuss the Monte-Carlo Glauber model. Next, in Sec.(IV), we present the
results. Finally, we conclude in Sec.(V). Throughout the paper, we use natural
units, $\hbar=c=k_{B}=\epsilon_{0}=\mu_{0}=1$.

##  II Formulations

As mentioned in the introduction, we consider the deceleration of charged
participants after they undergo binary collision. The electromagnetic fields
for a point particle with charge $Ze$ moving with velocity
$\bm{{\bm{{{\beta}}}}}=\frac{d\bf{r}}{dt}$ and a proper acceleration
$\bm{\dot{{\bm{\beta}}}}=\frac{d\bf{{\bm{\beta}}}}{dt}$ can be calculated at
position ${\bf{r}}_{\text{obs}}$ at time $t_{\text{obs}}$ from the well known
formula [50].

$\displaystyle e{\bf{B}}({\bf{r}}_{\text{obs}},t_{\text{obs}})$
$\displaystyle=-\mathcal{C}Z\alpha_{EM}\left[\frac{\hat{\bf{R}}\times{\bm{\beta}(t^{\prime})}}{\gamma^{2}k^{3}R^{2}}+\frac{(\hat{{\bf
R}}\cdot\dot{\bm{\beta}}(t^{\prime}))(\hat{\mathbf{R}}\times{\bm{\beta}}(t^{\prime}))+k\hat{{\bf
R}}\times\dot{{\bm{\beta}}}(t^{\prime})}{k^{3}R}\right]_{t^{\prime}},$ (1)
$\displaystyle e{\bf{E}}({\bf{r}}_{\text{obs}},t_{\text{obs}})$
$\displaystyle=\mathcal{C}Z\alpha_{EM}\left[\frac{\hat{{\bf{R}}}-{\bm{\beta}}(t^{\prime})}{\gamma^{2}k^{3}R^{2}}+\frac{\hat{\bm{R}}\times[(\hat{\bf{R}}-{\bm{\beta}}(t^{\prime}))\times\dot{{\bm{\beta}}}(t^{\prime})]}{k^{3}R}\right]_{t^{\prime}},$

where $e$ is electronic charge, the fields are all in units of $m_{\pi}^{2}$,
and $R$ is in fm. Here $\mathcal{C}=fm^{-2}/m_{\pi}^{2}\sim 2$ is a numerical
factor, $\alpha_{EM}=\frac{1}{137}$ is the fine structure constant, and $Z$ is
the atomic number of each nucleus (we consider symmetric collisions). The
right-hand side of the above expressions is evaluated at retarded time
$t^{\prime}$. The relation between $t^{\prime}$ and $t_{\text{obs}}$ is given
by

${}t^{\prime}+\sqrt{(x_{\text{obs}}-x^{\prime})^{2}+(y_{\text{obs}}-y^{\prime})^{2}+(z_{\text{obs}}-z^{\prime}(t^{\prime}))^{2}}=t_{\text{obs}}.$
(2)

The relative position
$\bf{R}(t^{\prime})=\bf{r}_{\text{obs}}-\bf{r^{\prime}}(t^{\prime})$, and the
unit vector along it is defined as $\hat{\bf{R}}=\frac{\vec{R}}{R}$, the
factor $k=1-\hat{\bf{R}}\cdot\bm{\beta}(t^{\prime})$, and
$\gamma=\frac{1}{\sqrt{1-\beta^{2}}}$ is the Lorentz factor.

To calculate the electromagnetic fields, we consider nucleons moving in a
straight line with a constant velocity
$\beta_{s_{NN}}=\left(1.0-\frac{4m^{2}}{s_{NN}}\right)^{1/2}$, $m$ is the mass
of the proton. Depending on whether they are participants or spectators, we
decelerate or let them continue moving with constant initial velocity for a
given $\sqrt{s_{NN}}$. Below, we discuss a step-by-step process for evaluating
the total electromagnetic field using an MC Glauber model and parameterized
form for deceleration.

  * •

We sample the positions of individual nucleons from the nuclear density
distribution of the Wood-Saxon type (the details of which are discussed
later). This will give us the initial positions $x^{\prime}_{0}(t^{\prime})$,
$y^{\prime}_{0}(t^{\prime})$ and $z^{\prime}_{0}(t^{\prime})$ of the nucleons
inside the right and left moving nucleus. We assume an Eikonal approximation
where, even after binary collisions, individual nucleons will continue moving
along the beam’s direction. So far, the calculation of the fields is done at
some retarded time $t^{\prime}$, and then we obtain the corresponding fields
at $t_{\text{obs}}$ using Eq.(2). Here, as we will be working with the
collision of individual nucleons tracking its trajectory, it is more
convenient to work in proper time $\tau$ where
$\tau=\sqrt{t^{2}-\mathbf{r}^{2}}$. Here, we choose $\tau=0$ at
$\mathbf{r}=0$, corresponding to $t=0$ when the centers of the two nuclei
overlap. In this scenario, individual nucleon-nucleon collisions can occur for
$\tau<0$ or $\tau>0$, assuming that just after the collision, the nucleons
that take part in the collision process would decelerate.

  * •

If the proton is a participant, we apply a deceleration with the following
parametrized form for the velocity profile as

$\displaystyle{}\beta(\tau)=\mathcal{A}\left[1-\tanh\left(\frac{\tau-\tau_{h}}{\Delta\tau}\right)\right],$
(3)

where, $\mathcal{A}=\frac{\beta_{s_{NN}}}{2}$, $\Delta\tau$ is a parameter
used to control the time interval for the deceleration, $\tau_{h}$ is a
parameter to control the time scale of the deceleration such that at
$\tau=\tau_{h}$ the initial velocity $\beta_{s_{NN}}$ is reduced to half of
its original value. Further we define a starting time $\tau_{s}$ for
individual nucleon-nucleon collision as the time when
$\beta_{s_{NN}}/\beta(\tau_{s})\sim 0.98$. As per our definition, $\tau_{s}$
could be positive or negative depending on the location of the participants
inside the colliding nucleus. The choice of the starting time of collision,
where velocity reduces to $98\%$ of the initial velocity, is reasonable.
Firstly, it helps us to handle the discontinuity in velocities at the onset of
deceleration. Moreover, from a physical perspective, this reduction can be
interpreted as arising from Coulombic repulsion or due to the composite
structure of the nucleons. We further note that the definition $\tau_{s}$ is
arbitrary. Still, it is a reasonable choice because we assume the nucleons in
the MC Glauber model are hard spheres with radius given by the inelastic
nucleon-nucleon cross section as given later in Eq.(5). A collision happens
when they touch each other.

The starting time for individual nucleon-nucleon collisions in a given event
is calculated based on their relative distances $\Delta
z=z_{target}-z_{projectile}$ at $\tau=0$, and considering straight line
trajectories with velocity $\beta_{s_{NN}}$.

Figure 1: (Color online) Parametrization of velocity $\beta$ ($\tau$) (solid
lines) and the corresponding $\frac{d\beta}{d\tau}$ (dashed lines) as a
function of $\tau$ for participants for $\Delta\tau$ = 1 fm (blue curves) and
3 fm (red curves).

In Fig.(1) we display the velocity profile $\beta(\tau)$ (solid lines) and the
proper acceleration $\frac{d\beta}{d\tau}$ vs $\tau$ (dashed lines) for
$\Delta\tau$ =1 fm (blue lines) and 3 fm (red lines) respectively.

  * •

In Eq.(1), EM fields are evaluated at retarded time; hence we need to find the
retarded time and corresponding positions of the nucleons from $\beta{(\tau)}$
using the relations

$\frac{dt^{\prime}(\tau)}{d\tau}=\frac{1}{\sqrt{1-\beta^{2}(\tau)}},\frac{dz^{\prime}(\tau)}{d\tau}=\frac{\beta(\tau)}{\sqrt{1-\beta^{2}(\tau)}}.$

  * •

Finally, we obtained the electromagnetic fields at observation point
${\bf{r}}_{\rm obs}\equiv(x,y,z)$ at present $t_{\text{obs}}$ from Eq.(1).
Calculating the electromagnetic field in an event-by-event case may result in
some nucleons being very close to the point of observation, making
$|\mathbf{R}|\approx 0$ leading to divergence for the fields. In practical
calculations, different regularisation schemes have been used, and consistent
results are obtained after taking the event average [4, 51, 1, 3]. To address
this issue, here we introduce a cutoff at $|\mathbf{R}|\approx$ 1 fm. This
implies that the fields associated with nucleons within a distance of
$|\mathbf{R}|=1$ fm are discarded.

In this work, we calculate EM fields from an ensemble of a thousand events for
a given collision centrality.

##  III Monte-Carlo Glauber

As mentioned earlier, we use the MC Glauber model [9] to calculate the nucleon
distributions, participants, spectators, and number of binary collisions for a
given $\sqrt{s_{NN}}$ and impact parameter, here we briefly discuss the
essential features and parameters used in our study. We sample the nucleon
positions inside a given nucleus from the corresponding Wood-Saxon density
distribution (assuming spherical symmetry):

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{1+e^{\frac{r-R}{a}}}.$ (4)

Where $\rho_{0}$ is the nucleon density in the center of the nucleus, R is the
radius of the nucleus, $a$ is the skin depth, and $r$ is the radial distance
from the center of the nucleus. For Au197 we use: $\rho_{0}=0.16$ fm-3,
$R=6.34$ fm, a = 0.54 fm. We calculate participant for a given nucleon-nucleon
inelastic cross section $\sigma_{NN}$ by considering individual nucleons as a
hard sphere; a collision takes place if the inter-nucleon transverse distance
$r_{\bot}=\sqrt{(x_{p}-x_{T})^{2}+(y_{p}-y_{T})^{2}}\leq r_{c}$, where

$\displaystyle r_{c}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\sigma_{NN}}{\pi}}.$ (5)

The ($x_{p}$, $y_{p}$) and ($x_{T}$, $y_{T}$) mentioned above are the
transverse positions of projectile and target nucleons respectively. The
experimentally measured values of $\sigma_{NN}$ are available for selected
energies, we fit the experimentally measured $\sigma_{NN}$ vs
$\sqrt{{s}_{NN}}$ with the following three parameters form,

$\displaystyle\sigma_{NN}^{f}$ $\displaystyle=$ $\displaystyle
A\left(\sqrt{{s}_{NN}}\right)^{B}+C.$ (6)

Here, from the fit we obtain the values $A=7.63$, $B=0.22$, $C=17.36$ with
$\chi^{2}\approx 0.05$. The parametric fit and experimental data points
(circles) are shown in Fig.(3). It is worthwhile noting that throughout the
rest of the paper, we assume that the trajectory of the participants is
governed by the Eq.(3) and each of the participants will eventually lose much
of its energy within a time interval $\Delta\tau$. However, the baryon usually
loses about one unit of rapidity in each collision, and it will take multiple
collisions before they lose a substantial amount of the initial energy [52,
53]. Such a realistic scenario of energy/rapidity loss is beyond the scope of
the present work and can be implemented in a future study.

Figure 2: (Color online) $\sigma_{NN}$ (nucleon-nucleon inelastic cross-
section) vs $\sqrt{s_{NN}}$ (circles) taken from [9], the red line is fit with
a three parameter function Eq.(6).

Figure 3: (Color online) Snapshot of the participants in a given Au+Au
collision event at $\tau$ $=$ 0 for $\sqrt{s_{NN}}$= 8 GeV and $b$=8 fm.

##  IV Results and Discussions

From now on, we denote ‘electronic charge times the fields’ simply as ‘Fields’
in our plots. We primarily focus on event-averaged values of the electric
${\bf{E}}\equiv(E_{x},E_{y},E_{z})\equiv E_{i}$ and magnetic
${\bf{B}}\equiv(B_{x},B_{y},B_{z})\equiv B_{i}$ field components to
investigate the effect of baryon stopping on the electromagnetic fields unless
stated otherwise. For the event-averaged case, we take ensemble of a thousand
events, and to focus on the event-by-event contribution of each field
component, we consider their absolute values so that the random phase
cancellation during the averaging could be avoided.

Figure 4: (Color online) Evolution of ${\bf E},{\bf B}$ at
$\textbf{r}_{\text{obs}}=(0,0,0)$ over time for $b=3$ (top) and $12$ fm
(bottom) at $\sqrt{s_{NN}}=4$ GeV, with $\Delta\tau=1$ fm.

Since the number of participants/collisions decreases monotonically with the
impact parameter/centrality of the collision, we expect the effects of baryon
stopping with deceleration on the fields is minimal for peripheral collisions.
Fig.(4) depicts the temporal evolution of $|E_{i}|$ and $|B_{i}|$ at the
center of the collision zone ${\bf r}_{\text{obs}}=(0,0,0)$, for
$\Delta\tau=1\,\text{fm}$ at $\sqrt{s_{NN}}=4\,\text{GeV}$ for two distinct
impact parameters: 3 fm and 12 fm, respectively. Upon comparison of both
plots, a notable distinction emerges, particularly evident in the more central
collision (at $b=3\,\text{fm}$), showcasing the effect of deceleration being
dominant at the most central collision which can be attributed to the
increased number of participants.

Figure 5: (Color online) Comparison of $|B_{x}|,|B_{y}|$ (top two panels), and
$|E_{x}|,|E_{z}|$ (bottom two panels) with (blue dashed lines) and without
deceleration (red solid lines) at $\sqrt{s_{NN}}=4$ GeV with $b$ = 3 fm at
$\textbf{r}_{\text{obs}}=(0,0,0)$.

In the presence of deceleration the magnitude of $|B_{x}|$, $|E_{x}|$,
$|E_{y}|$, and $|E_{z}|$ are enhanced in the presence of deceleration after
$\tau\sim$ 2 fm. Furthermore, electric fields seem to asymptotically reach a
constant value for $t_{\text{obs}}\gtrapprox 4\,\text{fm}$. This late-time
behavior arises from the dominance of Coulombic fields as deceleration drives
participant velocities towards non-relativistic limits, and they may
eventually come to rest. Additionally, a slight shift in the peak values of
$|B_{x}|$, $|B_{y}|$, $|E_{x}|$, and $|E_{y}|$ is noticeable in the top plot
compared to the bottom, primarily attributed to the velocity profile of
participants which start decelerating slightly before the collision, i.e.,
$\tau<0$. To see the effect of baryon stopping, we show the comparison of the
temporal evolution of the electromagnetic fields with and without baryon
stopping at $\sqrt{s_{NN}}=4$ GeV for $b=3$ fm in Fig.(5). The solid red lines
correspond to no stopping, and the blue dashed lines correspond to baryon
stopping. It is evident that $|E_{x}|$ and $|E_{z}|$ are significantly higher
for the stopping scenario at late times (after 3-4 fm). Whereas $|B_{y}|$ does
not show this type of asymptotic behavior at late times, as seen from the
second plot from the top in Fig.(5). This is understood as charges at rest
won’t give rise to any magnetic fields, unlike the Coulombic contribution to
the electric fields. The other components of the magnetic fields show similar
time dependence.

Figure 6: (Color online) Variation of magnetic fields with $\sqrt{s_{NN}}$ for
$b$= 3 (top panel), 12 fm (bottom panel) at ${\bf r}_{\text{obs}}$ = (0,0,0).

We found $|E_{z}|$ is most sensitive to deceleration and is enhanced
substantially after collisions for the baryon stopping scenario compared to
the other components of the electric fields. In case of no baryon stopping,
the peak value of event averaged electromagnetic field rises almost linearly
as a function of $\sqrt{s_{NN}}$ [4]. To investigate whether this approximate
linear dependency on $\sqrt{s_{NN}}$ still holds for baryon stopping scenario,
we plot $|B_{i}|$’s at $({\bf{r}}_{obs},t_{obs})=0$ as a function of
$\sqrt{s_{NN}}$ in Fig.(6). The top panel is for $b=3$ fm, and the bottom is
for $b=12$ fm. From the comparison of these two plots, a clear dependence is
observed as we go from central to peripheral collisions. However, the
approximate linear proportionality with $\sqrt{s_{NN}}$ holds for peripheral
collisions; the effect of baryon stopping seems to break the apparent
linearity. We found deceleration introduces a small quadratic dependence.
Fig.(7) on the other hand illustrates the impact parameter dependence of
$|E_{i}|,|B_{i}|$’s at $({\textbf{r}}_{\text{obs}},t_{\text{obs}})=({\bf
0},0)$ for baryon stopping (dashed lines) and without stopping (solid lines).
The change in the number of participants with impact parameters causes the
observed difference between the two cases.

Figure 7: (Color online) Variation of $B_{x}$ (red), $B_{y}$ (green) ,$B_{z}$
(blue), $E_{x}$ (cyan), $E_{y}$ (magenta), $E_{z}$ (yellow) with $b$ for
$\sqrt{s_{NN}}$ = 4 at $(t,{\bf{r}_{\text{obs}}})$ = $(0,{\bf{0}})$ with
(dashed lines) and without deceleration (solid lines).

The parameter $\Delta\tau$ used in Eq.(3) controls the timescale for the
deceleration of the participants. We study the effect of varying $\Delta\tau$
on the electromagnetic field in Fig.(8). The results for ($|B_{x}|$,
$|B_{y}|$) (top two panels) and ($|E_{x}|$ and $|E_{z}|$) (bottom two panels),
are shown for $\Delta\tau=1$ fm (solid red lines) and $\Delta\tau=3$ fm (solid
blue lines). We kept $b=3$ fm and $\sqrt{s_{NN}}=4$ GeV constant in all these
cases. We checked that $|E_{y}|$ is quite similar to that of $|E_{x}|$ and
hence not shown here. These results show a clear dependence of the field
strength and its time-evolution on $\Delta\tau$. For comparison, we also show
no deceleration results by dashed red lines.

Figure 8: (Color online) Comparison of the components of the fields for
$\Delta\tau$ = 1 (solid red lines), 3 fm (solid blue lines) and without
$\dot{\beta}$ (red dashed lines) at $\sqrt{s_{NN}}$ = 4 GeV and $b$ = 3 fm at
$\textbf{r}_{\text{obs}}$ = (0,0,0).

Naively, one would expect a large $\Delta\tau$ correspond to longer
deceleration time results in longer-living fields, but we do not expect any
dependency of the peak magnitude of the electromagnetic fields on $\Delta\tau$
for $\tau\leq 0$. However, in Fig.(8), we found an apparent deviation from
this expectation, particularly evident in the bottom panel. The peak magnitude
of $|E_{z}|$ appears to be higher for $\Delta\tau=1$ fm compared to
$\Delta\tau=3$ fm. This disparity primarily arises from the fact that in our
parametrization for velocity Eq.(3) a larger $\Delta\tau$ corresponds to a
smaller proper deceleration and vice versa and hence varying contribution in
the production of EM fields. Moreover, the lumpy charge distribution along the
longitudinal direction and the accumulation of these charges around the
observation point $\textbf{r}_{\text{obs}}=(0,0,0)$ after collision also
depend on $\Delta\tau$. The influence of $\Delta\tau$ seems to be more
prominent on the electric fields, particularly on $|E_{z}|$ shown in the
bottom panel of Fig.(8).

Figure 9: (Color online) Top panel: Fields at ${\bf r}_{\text{obs}}$ =
$(0,3,0)$ for $\sqrt{s_{NN}}$= 4 GeV and $\Delta\tau=$ 1 fm. Bottom panel:
same as top panel but for ${\bf r}_{\text{obs}}=(0,0,3)$.

Up until now, all the results shown were for ${\bf r}_{\text{obs}}$= (0,0,0).
In Fig.(9) we show the temporal variation of the fields at two different
observation locations ${\bf r}_{\text{obs}}$ = $(0,3,0)$ (top panel) and
$(0,0,3)$ (bottom panel) for $\sqrt{s_{NN}}$= 4 GeV and $\Delta\tau$ = 1 fm.
We found the temporal evolution of fields is almost identical for the
observation point located on the $y$ axis in the central transverse plane (top
panel) as what was observed at ${\bf r}_{\text{obs}}$ = $(0,0,0)$ (Fig.(4)).
The only exception is $|E_{y}|$, which is larger due to the the coherent
superposition from the target and projectile.

The passing of target and projectile nuclei through the observation point
${\bf r}_{\text{obs}}=(0,0,3)$ on the $z$ axis is expected to give rise to
double-peaked (symmetrically situated around $\tau=0$) structure of the
temporal evolution of field components which is apparent from the bottom panel
of Fig.(9). $|E_{z}|$ seems to dominate in this case compared to other
components. The asymmetry in the field values around $t_{\text{obs}}=0$ fm in
Fig.(9) arises due to the post-collision deceleration of nucleons.

Figure 10: (Color online) Fields in a randomly selected event. Top panel:
without deceleration at ${\bf r}_{\text{obs}}$ = $(0,0,0)$ for
$\sqrt{s_{NN}}$= 4 GeV and b= 3 fm. Bottom panel: same as top panel but with
deceleration and for the same event as above.

So far, all the results shown here have been obtained by taking averages of
the field components over many (thousand) events. We plotted all six field
components for a randomly chosen event for $\sqrt{s_{NN}}$= 4 GeV and b= 3 fm
in Fig.(10). The fact that field variations with time is non-trivial due to
the lumpy charge distribution and similar magnitude of the components of
electric and magnetic fields at ${\bf r}_{\text{obs}}=(0,0,0)$ is apparent
from the figure. The top panel corresponds to no deceleration, and the bottom
panel corresponds to deceleration case with $\Delta\tau$ = 1 fm.

##  V Conclusion

In this exploratory study, we use the Monte-Carlo Glauber model with post-
collision baryon stopping to investigate the vacuum space-time evolution of
electromagnetic fields in low-energy heavy-ion collisions. Several
observations are in order: firstly, upon incorporating baryon stopping via a
parameterized form of the velocity of the colliding nucleons, with the key
parameter being the time interval for deceleration ($\Delta\tau$), visible
effects are observed for $t_{\text{obs}}\geq 0$.

Secondly, as observed earlier for event-by-event calculations, due to the
quantum fluctuations in the nucleon positions, all components of the
electromagnetic fields become comparable in high-energy heavy-ion collisions
at ($\tau=0$); in the present study, we found temporal evolution of these
electromagnetic fields retain this trend before and after the collision.
Specific field components dominate for a given collision impact parameter only
when taking the event average. One of the novel finding of the present
investigation is that even without any medium effects the deceleration
enhances electric fields at late times compared to the case of no
deceleration; the effect of deceleration is most significant for the
longitudinal component of the electric fields at late times.

The presence of deceleration mostly reduces the strength of the magnetic
fields post-collisions. However, contrary to the other two components, we
found $B_{x}$ increases at late times compared to the scenario of zero
deceleration. We observed a slight shift of the peak position of EM fields
from $\tau=0$ for the baryon stopping scenario, perhaps akin to the particular
form of the parameterized velocity used in this study. For higher
$\sqrt{s_{NN}}$, when the nucleons move with almost constant velocity after
each binary collision, a linear proportionality of the peak value of magnetic
fields (at $\tau=0$) with $\sqrt{s_{NN}}$ is observed. However, in the
presence of deceleration, this approximate linearity seems to be broken
slightly at lower energies. Owing to the lower velocities of the colliding
nucleons for smaller $\sqrt{s_{NN}}$ the crossing time for the two nuclei
becomes longer and additionally, the fluctuating nucleon positions along the
longitudinal direction give rise to a non-trivial variation of the EM field
when measured on an event-by-event basis.

In future studies, several improvements are conceivable. The assumption that
nucleons, following collisions, come to an almost complete halt within a few
fm could be relaxed, and a more realistic scenario may entail nucleons
attaining a reduced velocity post-collision, with the possibility of further
energy loss upon subsequent collisions or maintaining this diminished velocity
depending on case-by-case. With this improvement, this model may match with
the experimentally measured net baryon density with rapidity, thereby
facilitating a more accurate estimation of $\Delta\tau$. During the conclusion
of our research, we encountered the article [54], where the authors also delve
into the realm of low-energy collisions, ranging from $\sqrt{s_{NN}}\sim 3$ to
10 GeV, and investigate the impact of baryon stopping utilizing a hadron
cascade model. Our study encompasses the effect of baryon stopping at the
initial state within the Glauber model, examining its influence on various
components of electromagnetic fields. In contrast, their research focuses on
the later stages of heavy-ion collisions, with the effect of stopping
considered at the final hadronic stage.

###### Acknowledgements.

A.P. acknowledges the CSIR-HRDG financial support. H.M. would like to thank
Sourendu Gupta and Sandeep Chatterjee for discussions. P.B. acknowledges
financial support from DAE project RIN 4001. V.R. acknowledges financial
support from SERB (CRG/2023/001309) , Government of India.

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