††thanks: Present address: Department of Mechanical Engineering, Stanford
University, Stanford, California 94305, USA<EMAIL_ADDRESS>
# Electron slingshot acceleration in relativistic preturbulent shocks explored
via emitted photon polarization

Zheng Gong<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Xiaofei Shen  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Karen
Z. Hatsagortsyan<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Christoph H. Keitel
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany

###### Abstract

Transient electron dynamics near the interface of counterstreaming plasmas at
the onset of a relativistic collisionless shock (RCS) is investigated using
particle-in-cell simulations. We identify a slingshot-like injection process
induced by the drifting electric field sustained by the flowing focus of
backwards-moving electrons, which is distinct from the well-known stochastic
acceleration. The flowing focus signifies the plasma kinetic transition from a
preturbulent laminar motion to a chaotic turbulence. We find a characteristic
correlation between the electron dynamics in the slingshot acceleration and
the photon emission features. In particular, the integrated radiation from the
RCS exhibits a counterintuitive non-monotonic dependence of the photon
polarization degree on the photon energy, which originates from a polarization
degradation of relatively high-energy photons emitted by the slingshot-
injected electrons. Our results demonstrate the potential of photon
polarization as an essential information source in exploring intricate
transient dynamics in RCSs with relevance for earth-based plasma and
astrophysical scenarios.

Plasma shocks are characterized by rapid steepening of a nonlinear wave, the
eventual overtaking by its rear part, and the irreversible energy transfer to
the surrounding particles [1, 2, 3]. They are of extensive interest in various
scenarios. In laboratory, a nonrelativistic shock can generate multi-MeV ions
in plasma-based accelerators [4, 5, 6, 7, 8, 9], improve the thermonuclear
gain of inertial confinement fusion [10, 11, 12, 13], and provide a platform
for investigating astrophysical phenomena [14, 15, 16, 17, 18, 19, 20]. For
astrophysics, the shock formed by supernova remnants offer plausible
mechanisms towards understanding the origin of TeV cosmic leptons [21, 22, 23,
24] and galactic PeVatrons [25], while RCSs are ubiquitous in pulsar wind
nebulae [26], active galactic nuclei [27], and gamma-ray bursts (GRBs) [28].
Recent observations suggest that the RCS prompted afterglow radiation
indicates a peculiar long GRB from the merger of a compact binary system [29,
30, 31, 32] rather than the core-collapse of massive stars [33], which
restimulates the research interest of relevant RCSs [34].

The RCS brewed from GRBs is prone to load with $e^{\pm}$ pairs since the
$\gamma$-ray photons ahead of the GRB ejecta turn into $e^{\pm}$ via
$\gamma$-$\gamma$ reaction [35, 36, 37]. Meanwhile, the magnetization level of
the unshocked interstellar medium is so low that the unmagnetized initial
condition is generally considered [38]. Here, the magnetized filamentation
turbulence, self-generated through the filamentation merging and magnetic loop
coalescence [39, 40, 41, 42, 43, 44, 45, 46], is crucial for determining
Weibel-mediated shock microstructures [47, 48, 49, 50, 51, 52, 53, 54], where
electrons, undergoing severe swirling and trace crossing, no longer travel in
a quasi-layer form. The electrons might experience stochastic acceleration,
akin to Fermi process [55, 56], which has been well recognized as sources of
energetic electrons in the universe [57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
67]. Previous studies of unmagnetized RCSs primarily focus on the electron
energization and equipartition between electrons and ions through stochastic
acceleration [68, 69, 70, 71, 72] where electrons scatter off the self-
generated turbulent magnetic structures [73, 74, 75]. However, it remains
largely unexplored how the plasma transits from the nonturbulent flow to
kinetic turbulence and how this transition impacts the acceleration and
radiation features in the RCS.

As a versatile information carrier of multi-messenger astrophysics [76, 77,
78], photon polarization is critical for measuring the magnetic configuration
nearby black holes [79] and crab nebulae [80]. Therefore, the question arises
whether the polarization feature of spontaneously emitted photons can be
employed to reveal the mechanism responsible for the turbulence transition in
a RCS.

In this letter, we investigate the transient electron dynamics in the
transition to turbulence nearby the counterstreaming interface of an
unmagnetized pair-loaded RCS precursor, which is potentially associated with
the outflow of GRBs. We employ PIC simulations to examine the photon emission
and observe an anomalous non-monotonic dependence (NMD) of the photon
polarization degree on the photon energy. We found that the NMD indicates a
specific mechanism of electron acceleration, which we term as slingshot
injection, caused by a drifting electric field due to the flowing focus of
backwards-moving electrons. Our slingshot model could be the essential reason
accounting for the long-term directed electron heating via electric fields
nearby a RCS precursor [81, 82, 83]. Utilizing Hamiltonian analyses, we
elucidate that the backwards-flowing focus marks the plasma transition to a
turbulent regime in the RCS, which in the electron’s transverse phase space is
exhibited as the change from the phase-locked to the phase-slipping dynamics.
The NMD photon properties stem from a polarization degradation of relatively
high-energy photons emitted by the slingshot-injected electrons. The
correlation among the NMD of photon polarization, the slingshot injection, and
the backwards-flowing focus emphasizes the importance of the transition region
to the turbulence in characterizing the acceleration and radiation in the RCS.

Figure 1: The dynamics of a counterstreaming RCS: The electron density $n_{e}$
at $t=80\pi/\omega_{pe}$ for (a) $\eta=0.4$ [84] and (b) $\eta=0.01$ [85],
where lines present the typical electron moving tendency with stars marking
the photon emission and the histograms display the spatial distribution of
emitted photons with $\omega_{ph}>10^{-2}\omega_{ph}^{m}$. (c)
$\left<\mathcal{Q}\right>$ and $\omega_{ph}dN_{ph}/d\omega_{ph}$ vs
$\omega_{ph}$. (d) $\left<\mathcal{Q}\right>$ and $dN_{ph}/d\theta_{ph}$ vs
$\theta_{ph}$.

We have carried out 2D simulations of counterstreaming RCSs, see Fig. 1. The
latter is initiated when a uniform plasma flow with a bulk Lorentz factor
$\gamma_{0}=50$, injected from the right side, is reflected from the left side
boundary, which adopts a reflection condition [58]. The periodic boundary
condition is set in the lateral direction. We consider the flow consisting of
electrons, positrons, and ions with the same drifting velocity and with the
number density of $n_{e0}$, $n_{p0}$, and $n_{i0}$, respectively. The charge
neutralization $n_{e0}=n_{p0}+Z_{i}n_{i0}$ is satisfied initially and the ion
with charge (mass) $Z_{i}=1$ ($m_{i}=1836m_{e}$) is use. The ratio $\eta\equiv
n_{i0}/(n_{i0}+n_{p0})\in(0.01,1)$ denotes the proportion of ions among the
whole positive charged particles. The simulation domain is
$200\lambda_{pe}\times 20\lambda_{pe}$ with resolution $\Delta x$=$\Delta
y$=$\lambda_{pe}/50$ and $\Delta t=0.95\Delta x/c$. Each cell is filled with
48 macro-particles for each species. Here,
$\omega_{pe}=(n_{e0}e^{2}/\varepsilon_{0}m_{e})^{1/2}$ ($\lambda_{pe}=2\pi
c/\omega_{pe}$) is the plasma frequency (skin depth), with the electron charge
(mass) $e$ ($m_{e}$), the vacuum permittivity $\varepsilon_{0}$, and the speed
of light $c$. The models of the photon polarization have been implemented in
the EPOCH code [86, 87]. Unless otherwise indicated, we discuss results from
the fiducial simulation with $\gamma_{0}=50$ and $\eta=0.4$.

The snapshot of the electron density $n_{e}$ in Fig. 1(a) exhibits that the
filamentation exclusively exists at the front of the RCS interface. Between
two adjacent filaments, an electron focusing point emerges, and following
that, two oblique density strips stretch out [see Fig. 2(b)]. Behind the
strips, the coherent filaments and focusing points disappear while the
turbulence shows up. A nontrivial thing is that the photons with energy
$\varepsilon_{ph}\equiv\hbar\omega_{ph}>10^{-2}\hbar\omega_{ph}^{m}$ are
primarily emitted by electrons nearby the interface, where
$\omega_{ph}^{m}\sim 10^{8}\omega_{pe}$ is the photon cut-off frequency and
$\hbar$ the Planck constant. In contrast, in the case of $\eta=0.01$, the
energetic photon emission predominantly occurs in the turbulent region [see
Fig. 1(b)], even though the preturbulent structures are extended to a larger
range.

The degree of photon’s linear polarization along the direction of the
electron’s transverse acceleration is characterized by the Stokes parameter
$\mathcal{Q}$ [88], formulated as [87]

$\displaystyle\mathcal{Q}=\frac{\varepsilon_{e}(\varepsilon_{e}-\varepsilon_{ph})K_{\frac{2}{3}}(\zeta)}{[\varepsilon_{e}^{2}+(\varepsilon_{e}-\varepsilon_{ph})^{2}]K_{\frac{2}{3}}(\zeta)-\varepsilon_{e}(\varepsilon_{e}-\varepsilon_{ph})\tilde{K}_{\frac{1}{3}}(\zeta)},$
(1)

where $K_{n}(\zeta)$ is the modified secondary Bessel function,
$\widetilde{K}_{1/3}(\zeta)=\int_{\zeta}^{\infty}K_{1/3}(z)\mathrm{d}z$,
$\zeta=2\varepsilon_{ph}/[3\chi_{e}(\varepsilon_{e}-\varepsilon_{ph})]$, and
$\varepsilon_{e}=\gamma_{e}m_{e}c^{2}$ the electron energy;
$\chi_{e}\equiv(e\hbar/m_{e}^{3}c^{4})|F_{\mu\nu}p^{\nu}|$ is the electron
quantum strong-field parameter with the field tensor $F_{\mu\nu}$, and the
electron four-momentum $p^{\nu}$. At $\chi_{e}\ll 0.1$,
$\partial\mathcal{Q}/\partial\varepsilon_{ph}>0$ predicted by Eq. (1)
manifests a monotonic dependence of $\mathcal{Q}$ on $\omega_{ph}$, because
for the higher-frequency radiation the formation length is shorter and the
preservation of the local polarization degree is improved. This monotonic
dependence is confirmed by the results of $\eta=0.01$ [see Fig. 1(c)(d)],
where electrons experience stochastic acceleration [59] and the photon
emission is isotropic in the angular space. However, for $\eta=0.4$ [see Fig.
1(c)(d)], the averaged polarization degree $\left<\mathcal{Q}\right>$ versus
$\omega_{ph}$ exhibits NMD, with a polarization dip
$\Delta\mathcal{Q}_{\omega}\approx 4.5\%$ and a bandwidth ratio
$\mathcal{B}_{\omega}\equiv\omega_{ph}^{m}/\omega_{ph}^{*}\sim 10^{3}$,
contradictory to the forementioned monotonic dependence. Here,
$\omega_{ph}^{*}$ is the local maximum point of the function
$\left<\mathcal{Q}\right>$ vs $\omega_{ph}$ [see Fig. 1(c)]. In the angular
distribution, $\left<\mathcal{Q}\right>$ has a polarization valley
$\Delta\mathcal{Q}_{\theta}\approx 11\%$ and the photon emission tends to be
more collimated within an emission angle $\theta_{ph}\lesssim 15^{\circ}$. The
features of the slingshot acceleration and the photon NMD are confirmed by 3D
PIC simulations [87].

To unveil the reason of the counterintuitive NMD, we focus on the electron
dynamics within the dashed box marked in Fig. 1(a). For the deflection of
backwards-moving electrons nearby the interface, the effective plasma density
approximates $\eta n_{e0}$ and the charge density has a sinusoidal profile
$\rho\sim|e|\eta n_{e0}\cos[k_{y}(y-y_{c})]$ with $k_{y}\sim\omega_{pe}/2c$
the periodic wave number and $y_{c}$ the relative central axis [87]. The self-
generated transverse electric and magnetic field is $E_{y}(y)=(|e|\eta
n_{e0}/\varepsilon_{0}k_{y})\sin[k_{y}(y-y_{c})]$ and $B_{z}(y)=cE_{y}$. As
justified by simulations, the energy exchange $d\gamma_{e}/dt$ is
insignificant and thus the transverse dynamics is described by
$\ddot{y}+(\Omega^{2}/k_{y})\sin[k_{y}(y-y_{c})]=0$, with $\Omega^{2}=2\eta
n_{e0}e^{2}/\varepsilon_{0}\gamma_{0}m_{e}$. Then the corresponding
Hamiltonian can be derived as [87]

$\displaystyle
H_{\perp}(y,\dot{y})=\frac{\Omega^{2}}{k_{y}^{2}}\cos[k_{y}(y-y_{c})]+\frac{1}{2}\dot{y}^{2}.$
(2)

Following $H_{\perp}(y,\dot{y})=H_{\perp}(y_{0},0)$, the electron transverse
motion is analyzed as

$\displaystyle
t=\frac{k_{y}}{\sqrt{2}\Omega}\int\frac{dy}{\sqrt{\cos[k_{y}(y_{0}-y_{c})]-\cos[k_{y}(y-y_{c})]}}.$
(3)

The trajectories predicted by Eq. (3) demonstrate that the backwards-moving
electrons would be focused into $y=y_{c}$ at a restoring time $t_{r}\sim
0.6\pi/\Omega$, as confirmed by simulation results [see Figs. 2(a)(b)].

Figure 2: (a) Schematic of the backwards-flowing focus with the predicted
electron trajectories [89]. The brown (purple) markers denote the magnetic
(electric) field direction and the green arrows present the slingshot-injected
electrons. (b) Zoom in on the dashed box marked in Fig. 1(a), where the red
(blue) lines represent the backward (forward) moving electrons [90]. (c)
Electron evolution in $(y,\dot{y})$ space with the blue dashed (red dotted)
lines contouring $H_{\perp}|_{\Omega\approx 0.1\omega_{pe}}$
($H_{\perp}|_{\Omega^{\prime}\approx 0.02\omega_{pe}}$) [91]. (d) Time
evolution of $\delta r$ ($\overline{\delta r}$) in grey (red).

After the backwards-flowing focus, the electrons start to transit from the
preturbulent motion to turbulence, interpreted as a shrinking of the
Hamiltonian’s separatrix. The separatrix $H_{\perp}(y,\dot{y})\equiv
H_{\perp}(y_{c},0)=\Omega^{2}/k_{y}^{2}$ divides the electron dynamics into
the confined phase-locked and the escaping phase-slippage regions. If the
magnetic field decreases with the equivalent restoring frequency reduced from
$\Omega$ to $\Omega^{\prime}$, the phase space volume encompassed by the
separatrix is shrunk from $H_{\perp}(y,\dot{y})<\Omega^{2}/k_{y}^{2}$ to
$H_{\perp}(y,\dot{y})<\Omega^{\prime 2}/k_{y}^{2}$. Thus, the electrons within
the region of $\Omega^{\prime
2}/k_{y}^{2}<H_{\perp}(y,\dot{y})<\Omega^{2}/k_{y}^{2}$ are released into the
phase-slippage region [see Fig. 2(c)]. The electron release breaks the
coherent filament structure and deteriorate the transverse inhomogeneity,
leading to the onset of the plasma turbulence.

The transition from the preturbulent flowing focus to the turbulence is
illustrated by the evolution of the particle separation [see Fig. 2(d)], where
$\delta r$ is the distance between an electron and its closest partner at the
beginning and $\overline{\delta r}$ refers to the averaged value. After the
focus at $\omega_{pe}t/2\pi\sim 45$, the signature of the chaotic dynamics
arises with $\overline{\delta r}\propto\exp{(\lambda_{l}\delta t)}$
characterized by the Lyapunov exponent $\lambda_{l}\approx
0.15\omega_{pe}/\pi$ [92]. The electrons exhibit a chaotic behavior during the
defocusing stage [93], where the decrease of the exerted magnetic field
$\overline{|B_{z}|}$ proves the shrinking of the Hamiltonian’s separatrix.
Later at $\omega_{pe}t/2\pi\sim 70$, $\overline{\delta r}\propto 0.2c\delta
t/\pi$ implies a drifting tendency because of the localized electrons prone to
occupy the whole interaction domain [94]. Eventually at
$\omega_{pe}t/2\pi>130$, $\overline{\delta r}\propto 9(\omega_{pe}\delta
t/2\pi)^{1/2}$ manifests the electrons’ random walk procedure [95, 96].

Figure 3: (a) Electric field $E_{x}$ with the transverse profile of $B_{z}$
and $E_{y}$. (b) $\delta l$ and $\left<E_{x}\right>$ vs $\eta$. (c)
Hamiltonian $\Pi_{\parallel}(\xi,p_{x})$ with the red arrows denoting the
moving tendency modified by the magnetic deflection. (d) Time-evolved electron
position, where the black lines profile $B_{z}$ and the red (blue) stars mark
the photon emission belong to the slingshot (stochastic) mechanism. Three
kinds of slingshot electrons are shown with the color of ‘A’ cyan, ‘B’ lime,
and ‘C’ magenta.

The flowing focus leads to a negative longitudinal electric field $E_{x}$ with
a scale length $\delta l$ [see Fig. 3(a)], favorable for injecting electrons
into the RCS. This injection resembles a slingshot, where the filaments serve
as the handhold, the backwards-moving electrons behave as the elastic string,
and the injected forwards-moving electrons are the projectiles [97]. The scale
length is calculated as $\delta l\sim
t_{r}c\approx\pi\sqrt{\gamma_{0}/\eta}(c/\omega_{pe})$. Given
$\nabla\cdot\bm{E}=\rho/\varepsilon_{0}$, the field strength is estimated as
$\left<E_{x}\right>\approx\pi\sqrt{\eta\gamma_{0}}m_{e}c\omega_{pe}/|e|$ [see
Fig. 3(b)]. The flowing focus successively occurs for the replenished
backwards-moving electrons and the field $E_{x}$ propagates with a velocity
$v_{x}\approx v_{d}=(1-1/\gamma_{0}^{2})^{1/2}$. In the interface’s co-moving
frame $\xi\equiv x-v_{d}t$, the electron’s longitudinal dynamics is determined
by the Hamiltonian
$\Pi_{\parallel}(\xi,p_{x})=-|e|\varphi(\xi)+c\sqrt{m_{e}^{2}c^{2}+p_{x}^{2}}-v_{d}p_{x}$
with $\varphi(\xi)=-\int E_{x}(\xi)d\xi$ [see Fig. 3(c)] [87]. Then the
injection threshold $p_{th}^{-}$ and the maximum achievable momentum
$p_{th}^{+}$, derived as [87]

$\displaystyle p_{th}^{+}\sim
2\gamma_{0}^{3}+\frac{3}{2}\gamma_{0}-\frac{2}{\gamma_{0}}\ \ \ \&\ \ \
p_{th}^{-}\sim-\frac{\gamma_{0}}{2}-\frac{1}{2\gamma_{0}}.$ (4)

Specifically, there are three types of slingshot-injected electrons [see Figs.
3] [98]. The ‘A’ electrons co-moving with $E_{x}$ get a pronounced energy gain
up to $\gamma_{e}\sim 10^{3}$. The initially backwards-moving ‘B’ electrons
are below the threshold, i.e. $p_{x}\sim-\gamma_{0}<p_{th}^{-}$, but they are
still injected because the magnetic deflection $\bm{v}\times\bm{B}$ leading to
an attractor effect in $(\xi,p_{x})$ space [99], which drags the electrons
towards the degraded Hamiltonian $\Pi_{\parallel}$ [see the red arrows in Fig.
3(c)] [87]. The ‘C’ electrons are trapped by the $E_{x}$ induced by assembling
two stretched-out density strips behind the flowing focus position. The
slingshot acceleration is distinguishable from the previously identified
stochastic acceleration, where electrons tend to be repetitively rebounded by
magnetic turbulence [73, 74, 75] and undergo Fermi-like stochastic
energization [68, 69, 70, 71, 72, 73]. Figure 3(d) manifests that the primary
contribution of photon emission nearby the preturbulent interface originates
from the slingshot electrons. The percentage of electrons undergoing a
slingshot process is $0.39\%$.

Figure 4: (a) Distribution of $W_{\parallel,\perp}$ at the emission moment.
(b) $\left<\mathcal{Q}\right>$ vs $\omega_{ph}$ for photons associated with
$W_{\parallel}/W_{\mathrm{t}}\lessgtr 0.75$, where the red line reproduces the
$\left<\mathcal{Q}\right>$ vs $\omega_{ph}$ in Fig. 1(c). (c) Time-evolved
electron energy spectral $\gamma_{e}^{2}dN/d\gamma_{e}$. The inset displays
the time-dependent electron maximum energy $\gamma_{e}^{m}$.

In the search for a criterion distinguishing between the slingshot and
stochastic electrons, we turn to the electron’s longitudinal and transverse
work $W_{\parallel,\perp}$ [see Fig. 4(a)], where
$W_{\parallel}=-\int|e|E_{x}dx$, $W_{\perp}=-\int|e|E_{y}dy$, and
$W_{\mathrm{t}}=W_{\parallel}+W_{\perp}$; the integrals are calculated from
the beginning to the photon emitting moment. The slingshot acceleration relies
on $E_{x}$ while the stochastic process is isotropic, meaning that the photon
emission associated with $W_{\parallel}/W_{\mathrm{t}}\rightarrow 1$
($W_{\parallel}/W_{\mathrm{t}}\rightarrow 0.5$) belong to the slingshot
(stochastic) mechanism [100]. Therefore, the condition of
$W_{\parallel}/W_{\mathrm{t}}\lessgtr 0.75$ is a reasonable criterion to
distinguish the photon emission from the stochastic or slingshot mechanism.
For the photons produced from the two mechanisms, both of their
$\left<\mathcal{Q}\right>$ vs $\omega_{ph}$ [in Fig. 4(b)] is monotonically
increasing as predicted by Eq.(1). However, the photon emission from the
slingshot is shifted to the higher frequency range compared with the
stochastic scenario due to the enhanced energy of slingshot electrons [see
Fig. 4(c)]. Therefore, the NMD of $\left<Q\right>$ vs $\omega_{ph}$ comes from
the combination between the high polarization degree stochastic photons and
the low polarization degree slingshot photons around $\omega_{ph}\sim
10^{6}\omega_{pe}$ [see Fig. 4(b)]. Nearby this frequency region, the emission
of both mechanisms contributes.

The maximum slingshot energy $\gamma_{sli}^{m}$ is approximated as
$\gamma_{sli}^{m}\sim|e|\left<E_{x}\right>\delta
t/m_{e}c\sim\pi\sqrt{\eta\gamma_{0}}\omega_{pe}\delta t$ when the electron
energy is far from the saturation $\gamma_{e}\ll p_{th}^{+}\sim 10^{6}$. The
energy gain of the stochastic process is estimated using the random walk model
[95]: $\gamma_{sto}^{m}\sim 0.5(\omega_{pe,0}\delta
t)^{1/2}\gamma_{0}^{3/4}\eta^{1/4}$ [87]. These estimates agree well with the
simulation results [Fig. 4(c)]. Following $\gamma_{sli}^{m}$ and
$\gamma_{sto}^{m}$, the photon cut-off frequency $\omega_{ph}^{m}$ for the
slingshot and stochastic mechanisms is predicted as
$\omega_{ph}^{m,sli}\sim\gamma_{sli}^{m\,2}B\propto\gamma_{0}^{3/2}\eta$ and
$\omega_{ph}^{m,sto}\sim\gamma_{sto}^{m\,2}B\propto\gamma_{0}^{2}\eta^{1/2}$
[see Fig. 5(a)] with the magnetic field strength $B\propto\gamma_{0}^{1/2}$.

Figure 5: (a) $\omega_{ph}^{m}$ vs $\eta$. (b) Dependence of
$\Delta\mathcal{Q}_{\omega}$ and $\mathcal{B}_{\omega}$ on $\eta$ and
$\gamma_{0}$, where the circles (triangles) refer to the electron dynamics
dominated by the slingshot (stochastic) mechanism.

Examining the NMD polarization features of the polarization dip
$\Delta\mathcal{Q}_{\omega}$ and the bandwidth $\mathcal{B}_{\omega}$, we
conclude that the high-frequency photon emission is dominated by the slingshot
mechanism due to fulfilling three criteria: i) the photon cut-off frequency
originating from slingshot electrons is much higher than via the stochastic
mechanism, i.e. $\omega_{ph}^{m,sli}\gg\omega_{ph}^{m,sto}$, reformulated as
$\eta\gtrsim\eta^{*}=0.01\gamma_{0}$; ii) the number of the slingshot injected
electron $N_{e}^{sli}\propto\left<E_{x}\right>$ should be larger than the most
energetic part of the stochastic electrons $N_{e}^{sto}\propto n_{pe0}$,
rearranged as $\eta\gtrsim\eta^{\dagger}\propto\gamma_{0}^{-1}$; iii) the
saturation of the slingshot acceleration should be higher than the stochastic
acceleration, i.e. $p_{th}^{+}\gtrsim\gamma_{sto}^{m}$, expressed as
$\gamma_{0}\gtrsim\gamma^{*}=2\eta^{1/9}$. The criteria of the slingshot
dominance predicted by
$\eta>\mathrm{max}\left\\{\eta^{*},\eta^{\dagger}\right\\}$ and
$\gamma>\gamma^{*}$ agrees well with the simulation results [see Fig. 5(b)].
The dependence of $\Delta\mathcal{Q}_{\omega}$ and $\mathcal{B}_{\omega}$ on
$\eta$ and $\gamma_{0}$ in Fig. 5(b) confirms that the NMD of the polarization
degree on photon energy is exclusively from the emission dominated by the
slingshot mechanism.

In conclusion, inspecting the origin of unexpected polarization features of
the photon radiation in the transient preturbulent RCS precursor, we have
identified the electron slingshot-like acceleration mechanism, distinct from
the well-known stochastic acceleration [68, 69, 70, 71, 72]. Our results have
implications for both laboratory and astrophysical phenomena. The identified
features of the transition region to turbulence, slingshot injection, and the
photon polarization dependence could be actualized in laboratory astrophysics
by using the combination of high-energy ion [101] and $e^{\pm}$ beams [102].
Moreover, the slingshot electrons, escaping from the preturbulent region to
enter the magnetic turbulent plasma [87], potentially behave as the pre-stage
injection for the subsequent Fermi acceleration in long-term evolved RCSs
[103]. Finally, the nontrivial photon polarization dynamics implies the
necessity of revising the retrieval model for astrophysical magnetic
configurations based on radiation features [104, 79, 105, 80].

###### Acknowledgements.

The original version of code EPOCH adapted here is funded by the UK EPSRC
grants EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/ M022463/1. The authors
would like to thank Laurent Gremillet, Anatoly Spitkovsky, and Dmitri Uzdensky
for the discussion regarding plasma stream instabilities, the initialization
of RCS in PIC simulations, and the undetermined composition of astrophysical
jets, respectively. The authors also thank the anonymous referees for their
useful comments and suggestions. Z. G. thanks Zhi-Qiu Huang for the gained
knowledge about the RCS generated following gamma-ray bursts.

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