# Polarisation signatures in radio
for inclined cosmic-ray induced air-shower identification

Simon Chiche<EMAIL_ADDRESS>Kumiko Kotera  Olivier Martineau-Huynh
Matias Tueros  Krijn D. de Vries  Sorbonne Université, CNRS, UMR 7095,
Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France
Sorbonne Université, Université Paris Diderot, Sorbonne Paris Cité,
CNRS,Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris,
France  IFLP - CCT La Plata - CONICET, Casilla de Correo 727 (1900) La Plata,
Argentina  Depto. de Fisica, Fac. de Cs. Ex., Universidad Nacional de La
Plata, Casilla de Coreo 67 (1900) La Plata, Argentina  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China  Vrije
Universiteit Brussel, Physics Department, Pleinlaan 2, 1050 Brussels, Belgium

(Received $\langle$date$\rangle$ / Accepted $\langle$date$\rangle$)

###### Abstract

Autonomous radio-detection, i.e., detection of air-showers with standalone
radio arrays, is one of the major technical challenges to overcome for the
next generation astroparticle detectors. In this context, we study
polarisation signatures of simulated radio signals to perform an
identification of the associated air-showers initiated by cosmic-rays and
neutrinos. We compare the two sources of radio emission (the charge excess and
geomagnetic) and show that the former is almost negligible for inclined
(zenith angle $>65\degree$) cosmic-ray air-showers. This provides an efficient
background rejection criterion at the DAQ level, based on the projection of
the total electric field along the direction of the local magnetic field. This
relevant quantity can be computed, —even in an online treatment— for antennas
measuring three orthogonal polarisations. Independently of the experimental
antenna layout, we estimate that assuming a random polarisation of noise
events, a rejection from $\approx 72\%$ (for a non favorable detector
location) to $\approx 93\%$ (for a favorable one) of the noise induced events
and a trigger efficiency of 86% (93%) with a $3\sigma$ ($5\sigma$) trigger
threshold level should be achievable. We also show that neutrino-induced
showers present a charge excess to geomagnetic signal ratio up to $\sim 10$
times higher than for cosmic ray showers. Although this characteristic makes
the identification of neutrino-induced showers challenging via the method
developed here, it provides an efficient criterion to perform an offline
discrimination between cosmic-ray and neutrino primaries. The stronger charge
excess emission will also help the reconstruction of air-shower parameters,
such as the core position.

###### keywords:

radio-detection, high-energy astroparticles, air-shower simulation,
polarisation

††journal: Astroparticle Physics

##  1 Introduction

Thanks to important experimental and theoretical progress in the past decade,
radio-detection was established as a robust and efficient technique to detect
and reconstruct the parameters of cosmic-ray induced air-showers (e.g., [1, 2]
for reviews). So far, most of these successful radio experiments have combined
radio antennas with particle detector arrays (e.g., AERA [3], CODALEMA [4],
LOFAR [5], Tunka-Rex [6]).

Autonomous radio-detection however, i.e., identifying radio signals with radio
antennas alone (without an external trigger such as particle detectors) is a
real challenge. The difficulties stem from the fact that transient pulses
induced by background sources (radio relays, planes, storms…) are far more
numerous – by a factor $10^{4}$ at least outside polar areas [7] – than those
induced by astroparticles. This requires an efficient rejection process
performed as early as possible in the experimental chain. A handful of
experiments have already shown promising results towards autonomous radio
detection [8, 9, 10, 11, 12], yet more robust methods of background rejection
have to be developed to achieve higher efficiency. Autonomous radio-detection
is of primary importance for the next generation of radio experiments with a
large number of antennas, such as GRAND [13].

Most of the experiments cited above have mostly detected cosmic-ray air-
showers with zenith angles lower than 60°. Very inclined air-showers were also
studied by ANITA and AERA for example and present different properties due to
their longer propagation in a thiner atmosphere. The detection and study of
such events has gained momentum, with the conception of large-scale and sparse
radio arrays (e.g., GRAND [13] and the Auger Radio Upgrade [14]), endeavouring
to sample the large radio footprints of inclined showers. Such inclined
showers could be generated by ultra-high-energy tau neutrinos with Earth-
skimming trajectories.

In this study, we will investigate the specific polarisation features of the
radio signal and establish an identification procedure which provides an
instrumental way to address the challenge of autonomous radio-detection for
cosmic-ray induced inclined air-showers. Additionally, we will also show how
polarisation signatures could enable the discrimination of cosmic-ray and
neutrino induced air-showers.

The polarisation is defined as the direction of the electric field vector at a
given instant. As the radiation from particles in the shower adds up
coherently, we expect a highly polarised signal [15, 16]. The emission
mechanisms that contribute to the polarisation are well known and were overly
described by Classical Electrodynamics calculations [17, 18, 19]. Polarisation
can directly be measured with radio antennas with two or three arms, that
would measure the different components of the electric field. Polarisation is
known to be an efficient tool to perform background rejection [15, 20, 21, 22,
23], we aim here at quantifying quantitatively how efficient this tool can be
for the detection of very inclined air-showers.

We recall the main mechanisms responsible for the radio emission in air-
showers in Section 2. In Sections 3 and 4, we reconstruct the polarisation
from numerical simulations of air-showers. Then in Section 5, we use the
reconstructed polarisation to highlight the characteristics that we exploit to
perform autonomous radio detection. We present our results in Section 6, first
on a star-shape detector layout and then on an experimental layout and define
criteria for trigger at the DAQ level. Finally, in Section 7, we examine the
case of neutrino induced air-showers.

##  2 Radio emission from extensive air-showers

The radio emission from extensive air-showers is mainly the superposition of
two mechanisms with different linear polarisation patterns: the geomagnetic
emission and the charge excess emission.

###  2.1 Geomagnetic emission

In an Extensive Air-Shower (EAS), particles of charge $q$ are deflected by the
geomagnetic field $\mathbf{B}$ due to the Lorentz force:
$\mathbf{F}=q\mathbf{v}\times\mathbf{B}$ where $\mathbf{v}$ is the particle
velocity. The deflection in opposite direction of the lightest particles
mostly, i.e., electrons and positrons in combination with an effective
friction force due to multiple scattering results in a net transverse current
in the cascade front. This current varies over time as the number of electrons
and positrons changes during shower development. This induces a linearly
polarised radio emission orthogonal to the direction of the Earth magnetic
field and the shower axis, i.e., along $-\mathbf{v}\times\mathbf{B}$ as can be
seen on the left panel of Fig. 1.

The geomagnetic electric field amplitude scales with $\approx vB\sin{\alpha}$,
with $\alpha$ the angle between the shower axis and the local Earth magnetic
field direction. It also increases with the number of charged particles
(coherence effect) and with the mean free path of electrons before they loose
their energy during strong inelastic processes like Bremsstrahlung. This
implies that the amplitude of the geomagnetic emission is expected to decrease
with increasing density of the medium.

Figure 1: Total (geomagnetic + charge excess) electric field amplitude (color
scale) and direction (arrows) of the geomagnetic (left) and charge excess
(right) emissions, for a shower with primary particle energy $E=0.68\,$EeV and
zenith angle $\theta=53$°, represented in the plane perpendicular to the
shower axis (${\bf e}_{\mathbf{v}\times\mathbf{B}}$, ${\bf
e}_{\mathbf{v}\times(\mathbf{v}\times\mathbf{B})}$). A small number of
antennas is also generated outside of the star-shape pattern in our
simulations, for cross checks.

###  2.2 Charge excess or Askaryan emission

As the shower propagates through the atmosphere, electrons from the air atoms
are struck by high energy shower particles. This results in an accumulation of
negative charges in the shower front, mostly close to the intersection of the
shower axis and the shower front, the so-called shower core. The net negative
charges can be assimilated to a point charge at the shower core comparatively
to the longitudinal extent of the shower. This results in an electric field
oriented toward the core. As illustrated in the right panel of Figure 1, the
charge excess emission leads to a linear radial polarisation in the shower
plane $({\bf e}_{\mathbf{v}\times\mathbf{B}},{\bf
e}_{\mathbf{v}\times(\mathbf{v}\times\mathbf{B})})$, defined as the plane
perpendicular to the shower axis, with ${\bf e}_{\mathbf{v}\times\mathbf{B}}$
and ${\bf e}_{\mathbf{v}\times(\mathbf{v}\times\mathbf{B})}$, unit vectors
along the $\mathbf{v}\times\mathbf{B}$ and
$\mathbf{v}\times(\mathbf{v}\times\mathbf{B})$ directions respectively.

The charge excess electric field amplitude increases with the density of the
medium in which the shower develops [2]. It accounts for about $10\%$ of the
geomagnetic amplitude for vertical air-showers after correction of
$\sin{\alpha}$.

##  3 ZHAireS simulations and signal processing

This study is based on the analysis of data obtained with the ZHAireS code.
ZHAireS is a Monte-Carlo simulation of the microscopic emission by individual
particles in the shower [24, 25]. These simulations can generate showers with
various primary particles (proton, iron, gamma…), various directions (azimuth
and zenith angles) and various energies. We used the 19.04.00 version of Aires
with SYBILL2.3 as hadronic model combined with the 1.0.28 version of the
ZHAireS extension. All the simulations performed for this article were done
using a magnetic field inclination of $60.79\degree$, while the observer
altitude level for star-shape simulations was fixed at 1080 m above sea level.

The outputs of ZHAireS simulations are the electric field time-traces $E_{\rm
x}(t)$, $E_{\rm y}(t)$, $E_{\rm z}(t)$ measured at each antenna position,
where we choose $x$ = South-North, $y$ = West-East and $z$ = up. Figure 2
(left panel) shows an example of $E_{\rm y}(t)$ received at one given antenna
and Figure 3 the total peak-to-peak amplitude for a star-shape layout of
antennas. The trace is sampled at a 0.5 ns rate. The layout here is centered
on the air-shower core position, it appears that the signal is peaked on a
circular region which corresponds to antennas on the Cerenkov cone of the
radio signal.

Figure 2: (Left) Example of a raw trace along the $y$-axis (East-West) at a
given antenna, for a shower with primary energy $E=0.68\,$EeV and zenith angle
$\theta=53\degree$. (Right) same trace after adding a gaussian stationary
noise of $RMS=20\mu$V/m.  Figure 3: Total electric field peak-to-peak in the
geographic plane for a star-shape antenna layout considering a proton induced
shower with primary particle energy $E=0.68\,$EeV and zenith angle
$\theta=53\degree$.

To reconstruct the polarisation, we model the experimental measurement of the
radio signal by the antennas. For this purpose, we first add a background
noise to the traces to model the stationary noise coming from the Milky Way.
This is done by adding a Gaussian stationary noise of root mean square (RMS)
20 $\mu$V/m, computed following Eq. 2 of [26], where the sky brightness
follows the frequency dependency computed from Galactic emission measurements.
This is illustrated in the right panel of Fig. 2. We consider that there is no
correlation between the three components ($x$, $y$ and $z$) of the noise and
therefore add a different noise realisation to each channel of the traces. It
should be noted that ideally this study should be repeated on actual noise
measurements for example at the GRAND site. However, with no such data
available, we performed an idealized study using randomly polarized Gaussian
noise.

We then apply a Butterworth filter of order 5 in the frequency range of the
GRAND experiment ($50-200$ MHz). Finally, we sample the traces at a 2 ns
period to derive the experimental measurement of the radio signal. The
filtering and the sampling processes are illustrated in Fig. 4.

Figure 4: Traces from Fig. 2 after filtering (solid line), and sampling
(orange dots).

Once the traces are fully processed (noise, filtering, sampling), we can
reconstruct the polarisation. As we aim to present general features of the
polarisation independent of the experiment, we do not take into account any
specific antenna response in this treatment. However, to account for the
calibration error when reconstructing the polarisation, we apply a factor
randomly drawn in a Gaussian distribution of mean value 1.0 and standard
deviation 0.1 for each antenna. An identical value is chosen for the $x$ and
$y$ axes, and an independent one for the $z$ axis, thus accounting for the
different antenna response and wave reflection for horizontal and vertical
polarisations.

##  4 Polarisation reconstruction

We detail here the steps to reconstruct the polarisation from the time traces
presented in the previous section. The analysis presented in this Section and
in Section 5 follows mainly what was done by the Auger collaboration in [27],
but is presented here with our own set of simulations for self-consistency. We
use a standard method based on the Stokes parameters as detailed in Ref. [22,
28], and work in the shower plane (${\bf e}_{\mathbf{v}\times\mathbf{B}}$,
${\bf e}_{\mathbf{v}\times(\mathbf{v}\times\mathbf{B})}$), the relevant
framework to study polarisation.

###  4.1 Stokes parameters

The Stokes parameters are four parameters that allow to completely describe
the polarisation of a temporal signal in a plane. For temporal values of the
electric field $E_{v\times B}(t)$, $E_{v\times(v\times B)}(t)$ given in a
plane, the Stokes parameters are defined as follows:

$\displaystyle I$ $\displaystyle=$
$\displaystyle\dfrac{1}{n}\sum_{i=1}^{n}x_{i}^{2}+\widehat{x}_{i}^{2}+y_{i}^{2}+\widehat{y}_{i}^{2}$
(1) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\dfrac{1}{n}\sum_{i=1}^{n}x_{i}^{2}+\widehat{x}_{i}^{2}-y_{i}^{2}-\widehat{y}_{i}^{2}$
(2) $\displaystyle U$ $\displaystyle=$
$\displaystyle\dfrac{2}{n}\sum_{i=1}^{n}x_{i}{y}_{i}+\widehat{x}_{i}\widehat{y}_{i}$
(3) $\displaystyle V$ $\displaystyle=$
$\displaystyle\dfrac{2}{n}\sum_{i=1}^{n}\widehat{x}_{i}{y}_{i}-{x}_{i}\widehat{y}_{i}\
,$ (4)

where $n$ corresponds to the number of time sample used for computation,
$x_{i}$ and $y_{i}$ correspond to the values of the traces $E_{v\times B}(t)$,
$E_{v\times(v\times B)}(t)$ at an instant $i$ and $\widehat{x}_{i}$ and
$\widehat{y}_{i}$ to the imaginary part of the traces obtained by extending
the traces in the complex domain using the Hilbert transform.

The Stokes parameter $I$ is related to the total intensity of the radio signal
and can hence be used to define the time window for the computation of Stokes
parameters. Instead of averaging $I$ over the full time sample as in Eq. 1, we
can construct a time series and derive the intensity for each individual
sample. We then define the time window over which the traces and the Stokes
parameters are averaged as the FWHM111The full width half maximum corresponds
to the with of the peak at half of its maximum. of the time series as was done
by Ref. [28] and illustrated in Figure 5 for a given trace. This time window
is designed to maximise the signal-to-noise ratio of the signal.

Figure 5: Example of time window defined using the FWHM of the peak for a
shower with $E=0.68\,$EeV and $\theta=53\degree$.

###  4.2 Polarisation in the shower plane

For any elliptical polarisation, the Stokes parameters $U$ and $Q$ can be
expressed as a function of the ellipticity angles $\Psi$ and $\chi$ as
$Q=I\cos{2\psi}\cos{2\chi}$ and $U=I\cos{2\psi}\sin{2\chi}$. In the particular
case of a linear polarisation and for Stokes parameters computed in the shower
plane we have $\chi=\phi_{\rm p}$, where $\phi_{\rm p}$ is the polarisation
angle, i.e., the angle between the direction of the polarisation and the ${\bf
v}\times{\bf B}$ direction, as illustrated in Fig. 7. Then, once we have
averaged the 4 Stokes parameters over the time window displayed in Fig. 5 the
polarisation angle reads

$\phi_{\rm p}=0.5\arctan{\frac{U}{Q}}\ .$ (5)

Assuming the total polarisation to be linear, we can finally express the
components of the polarisation in the shower plane (directly related to the
direction of the electric field vector) as follows:

$\displaystyle E_{v\times B}$ $\displaystyle=$
$\displaystyle\sqrt{I}\cos{\phi_{\rm p}}\ ,$ (6) $\displaystyle
E_{v\times(v\times B)}$ $\displaystyle=$ $\displaystyle\sqrt{I}\sin{\phi_{\rm
p}}\ .$ (7)

The reconstructed polarisation in the shower plane is represented in Fig. 6
for a given shower. One can observe that the total polarisation is
predominantly aligned with $-\mathbf{v\times B}$ as expected for a dominant
geomagnetic emission.

Figure 6: Total polarisation inferred from Stokes parameters in the shower
plane for a shower with primary particle energy $E=0.68\,$EeV and zenith angle
$\theta=53\degree$.

##  5 Characteristics of the charge excess to geomagnetic ratio

Exploiting polarisation patterns of both emission mechanisms (a geomagnetic
contribution along $-\mathbf{v}\times\mathbf{B}$ and a radial charge excess),
we can use the total polarisation to infer the amplitude of each mechanism. In
particular, as the field along $\mathbf{v}\times(\mathbf{v}\times\mathbf{B})$
corresponds to charge excess only, knowing the direction of the charge excess
emission (radial in the shower plane), we can infer its norm. The field along
$\mathbf{v}\times\mathbf{B}$ however, is the result of the interference
between the charge excess and geomagnetic emission. Therefore after inferring
the norm of the charge excess we can subtract its contribution to the
$\mathbf{v}\times\mathbf{B}$ component of the electric field to derive the
norm of the geomagnetic emission. Hence, amplitudes of both mechanisms can be
derived from the following expressions [21, 27]

$\displaystyle E_{\rm ce}$ $\displaystyle=$ $\displaystyle\frac{E_{\rm
v\times(v\times B)}}{|\sin{\phi_{\rm obs}}|}\ ,$ (8) $\displaystyle E_{\rm
geo}$ $\displaystyle=$ $\displaystyle E_{\rm v\times B}-E_{\rm v\times(v\times
B)}\frac{\cos{\phi_{\rm obs}}}{|\sin{\phi_{\rm obs}}|}\ ,$ (9)

where $\phi_{\rm obs}$ is the angle between the antenna position and the
$\mathbf{v}\times\mathbf{B}$ axis. One should note that these equations are
not applicable for the horizontal baseline of antennas (along the
$\mathbf{v}\times\mathbf{B}$ axis) as along this baseline, the charge excess
has no component along $\mathbf{v}\times(\mathbf{v}\times\mathbf{B})$.

Figure 7: Representation of $\phi_{\rm p}$ and $\phi_{\rm obs}$ in the shower-
plane frame $(O,{\bf e}_{{\bf v}\times{\bf B}},{\bf e}_{{\bf v}\times{\bf
v}\times{\bf B}})$, with $O$ the shower core position, ${\bf B}$ the
geomagnetic field and ${\bf k}$ the shower axis. ${\bf E}$ is the electric
field at observer position $P$.

From the reconstructed components of the charge excess and geomagnetic
emission, we can then infer the charge excess to geomagnetic ratio defined as
follows:

$a\equiv\sin\alpha\,\frac{E_{\rm ce}}{E_{\rm geo}}\ ,$ (10)

where we correct the geomagnetic emission from $\sin{\alpha}$. This is done in
order to infer general results that do not depend on the azimuth of the
shower.

In the following sections, we will present the dependencies of the $a$ ratio
with shower parameters. Following what was presented in [23], we will study
how this ratio provides an efficient way to identify signatures from inclined
cosmic-ray air-showers and allows to impose constraints on the trigger
condition to perform a strong background rejection. We will also demonstrate
that it can help discriminate between cosmic-ray and neutrino induced showers.

###  5.1 Evolution with angular distance

Figure 8: Evolution with angular distance $\omega$ of the charge excess to
geomagnetic ratio as reconstructed with Eqs. (9-8) for a simulation of a
proton-induced shower with energy $E=0.68\,$EeV and zenith angle
$\theta=53\degree$ on a star-shape layout.

In Figure 8, we represent the charge excess to geomagnetic ratio for a given
simulation without noise on a star-shape layout as a function of $\omega$, the
angular deviation to the shower axis measured from $X_{\rm max}$, i.e.
($\widehat{\bf{u_{v}},\bf{u_{\rm antenna}}}$), with $\bf{u_{v}}$ the unit
vector related to the direction of the shower and $\bf{u_{\rm antenna}}$ the
unit vector that goes from $X_{\rm max}$ to a given antenna. Using this angle
is particularly convenient as shower properties are expected to be similar for
antennas located at the same $\omega$. Indeed, the amplitude pattern of the
radio signal in the shower plane becomes invariant by translation along the
shower axis with this representation. We can observe a quasi-linear increase
of the charge excess to geomagnetic ratio with $\omega$. This linear increase
with $\omega$ —or similarly with distance to the shower core— was already
discussed in [29] or [15] and can be justified considering simple scaling laws
within the analytical framework of Ref. [16]. As the charge excess contributes
to the scalar potential and the geomagnetic to the vector potential [18], the
related electric fields read

$\displaystyle E_{\rm geo}$ $\displaystyle\approx$ $\displaystyle\frac{{\rm
d}t_{r}}{{\rm d}t}\frac{{\rm d}A^{\rm geo}}{{\rm d}t_{r}}\ ,$ (11)
$\displaystyle E_{\rm ce}$ $\displaystyle\approx$ $\displaystyle\frac{{\rm
d}t_{r}}{{\rm d}x}\frac{{\rm d}A_{\rm ce}}{{\rm d}t_{r}}\ .$ (12)

where we introduced $A^{\rm ce}$ and $A^{\rm geo}$ the potentials related
respectively to the charge excess and the geomagnetic emissions and $t_{r}$,
the retarded time. Then, from the derivatives of $t_{r}$ [18] the charge
excess to geomagnetic ratio yields

$\displaystyle a\equiv\sin{\alpha}\frac{E_{\rm ce}}{E_{\rm geo}}$
$\displaystyle\approx$ $\displaystyle\sin{\alpha}\frac{C_{x}}{\langle
v_{d}\rangle}\frac{{\rm d}t_{r}}{{\rm d}x}\left(\frac{{\rm d}t_{r}}{{\rm
d}t}\right)^{-1}$ (13) $\displaystyle\approx$
$\displaystyle\sin{\alpha}\frac{C_{x}}{\langle v_{d}\rangle}\frac{x}{R}$ (14)
$\displaystyle\approx$ $\displaystyle\sin{\alpha}\frac{C_{x}(\rho)}{\langle
v_{d}(\rho)\rangle}\sin{\omega}\ ,$ (15)

with $C_{x}$ being the fraction of particles that contributes to the charge
excess with respect to the total number of positrons and electrons, $R$ the
distance between the source and the observer, $\langle v_{d}\rangle$ the mean
drift velocity of the electrons in the air-shower plasma (in units of $c$) and
$\alpha$ the geomagnetic angle. The dependency with $\omega$ indicates that
for the radio emission of a given air-shower, the charge excess to geomagnetic
ratio will not be spatially uniform. Usually for $\omega>4\degree$ the signal
amplitude is too low to be measured, then restricting to $\omega$ values below
4$\degree$ we can get the approximation $\sin{\omega}\approx\omega$ confirming
the linear increase of the charge excess to geomagnetic ratio with $\omega$
and hence with distance to the shower core.

Another interesting feature in Fig. 8 is the presence of a peak in the ratio
around $\omega$ = $1$°. This peak was also noticed by [19, 30] but its origin
remains debated [30, 29]. Cerenkov-like effects could be linked to this peak.
For example, if the charge excess emission peaks deeper in the atmosphere than
the geomagnetic emission, its Cerenkov cone is expected to be closer to the
shower axis than the Cerenkov cone related to the geomagnetic emission due to
density effects. Consequently, we expect an increase in the ratio for an
$\omega$ angle corresponding to the maximum of emission of the charge excess
and then a decrease as we reach the the maximum related to the geomagnetic
contribution, resulting in a peak for the ratio.

In Eq. (15), we also highlighted the dependency of the charge excess to
geomagnetic ratio with the density of the medium $\rho$ as it was also well
highlighted and parametrized from CoREAS simulations in [31]. This dependency
indicates that the charge excess to geomagnetic ratio should also vary with
the geometry of the showers as inclined showers for example are expected to
develop in thinner atmosphere than vertical air-showers. Particularly, the
mean drift velocity $\langle v_{d}\rangle$ is expected to decrease with
increasing density as the mean free path of electrons and positrons before
collision with air atoms will be shorter. At the same time, for a denser
medium we expect larger values of $C_{x}$ as there will be more air-atoms
colliding with the air-shower, resulting in a larger number of electrons in
the shower front. This feature will specifically be detailed in the next
section.

###  5.2 Dependency of the $a$ ratio on primary energy and zenith inclination

The dependency of the $a$-ratio with zenith angles and energy was discussed in
[31]. For the purpose of self-consistency, we derive these dependencies from
our set of simulations. We first consider a set of $\sim 5\,000$ simulations
of proton- or iron-induced showers with various energies and directions (14
bins of energy and 5 bins of zenith angles corresponding to 75 simulations for
each couple of energy and zenith, see Table 1) and apply the method presented
in Section 4 to reconstruct the charge excess and geomagnetic energy. We do so
by integrating the fluence of antennas along the $\mathbf{v\times(v\times B)}$
axis, where both emissions are uncorrelated following [32] and assuming a
radial symmetry of the radio signal in the shower plane. This was first done
for raw, unfiltered traces, i.e., traces before any processing detailed in
Section 3. In Fig. 9, we represent the ratio between the charge excess and
geomagnetic energies as a function of the primary energy and zenith angle for
simulations with proton and iron primaries.

| Energy
---
(EeV)
0.20 | 0.63 | 1.26 | 2.00 | 3.98 |  |  |
| Zenith $\theta$
---
(Deg.)
38.2 | 49.1 | 57.0 | 63.0 | 67.8 | 71.6 | 74.8 | 77.4
| 79.5 | 81.3 | 82.7 | 83.9 | 85.0 | 85.8 | 86.5 | 87.1
| Azimuth $\phi$
---
(Deg.)
0 | 90 | 180 |  |  |  |  |
Table 1: Lists of the primary particle energies $E$, zenith angles $\theta$ and azimuth angles $\phi$ of the arrival direction considered in Section 5.2. | Energy E
---
(EeV)
0.020 | 0.025 | 0.032 | 0.040 | 0.050 | 0.063 | 0.079 | 0.100 | 0.126 | 0.158 | 0.200 | 0.251
| 0.316 | 0.398 | 0.501 | 0.631 | 0.794 | 1.000 | 1.259 | 1.585 | 1.995 | 2.511 | 3.162 | 3.981
| Zenith $\theta$
---
(Deg.)
81.26 | 82.72 | 83.94 | 84.95 | 85.80 | 86.50 | 87.08 |  |  |  |  |
| Azimuth $\phi$
---
(Deg.)
0 | 90 | 180 |  |  |  |  |  |  |  |  |
Table 2: Lists of the primary particle energies $E$, zenith angles $\theta$ and azimuth angles $\phi$ of the arrival direction considered in Section 6.1. | Energy
---
(EeV)
0.63 | 0.79 | 1.00 | 1.25 | 1.58 | 1.99 | 2.51 | 3.16 | 3.98
| Zenith $\theta$
---
(Deg.)
81.26 | 82.72 | 83.94 | 84.95 | 85.8 | 86.5 | 87.08 |  |
| Azimuth $\phi$
---
(Deg.)
0 | 90 | 180 | 270 |  |  |  |  |
Table 3: Lists of the primary particle energies $E$, zenith angles $\theta$
and azimuth angles $\phi$ of the arrival direction considered in Section 6.2.

Figure 9: Ratio between the charge excess to geomagnetic radiated energies
corrected from the geomagnetic angle for raw traces of proton- (left) and
iron- (right) induced showers as a function of the zenith angle and the energy
of the primary particle. From a set of $\sim 5\,000$ simulations, we average
the $a$-ratio of simulations having the same couple of zenith angle and
energy, the error bars correspond to the RMS of each distribution. The
radiated energy of both emission are derived by integrating the fluence of
antennas along the ${\bf v\times(v\times B)}$ axis and assuming a radial
symmetry of the radio signal.

From these figures, it appears clearly that there is a mild correlation with
the energy of the primary particle and a strong one with zenith angle. This
can be explained by density arguments:

Regarding the correlation with energy, it is expected that the $X_{\rm max}$
position of the air-shower should scale logarithmically with the energy of the
primary [33]. This implies that the density at $X_{\rm max}$ should also
increase with increasing energy. Hence considering that most of the radio
emission occurs at the maximum air-shower development depth $X_{\rm max}$, it
appears in Eq. (15) that a higher density at $X_{\rm max}$ will result in a
stronger charge excess, due to an enhanced accumulation of negative charges
(increase of $C_{x}$) and a weaker geomagnetic emission (decrease of $\langle
v_{d}\rangle$).

Regarding the dependency with the zenith angle, the $X_{\rm max}$ position is
shallower for inclined showers, hence the charge excess to geomagnetic ratio
lower, following Eq. (15).

Moreover, the shower maximum $X_{\rm max}$ should be shallower for iron than
proton nuclei as the higher mass number increases the interaction probability
[1]. This implies that the charge excess to geomagnetic ratio is expected to
be lower for iron-induced showers. This is confirmed with simulations as
illustrated in Fig. 9. To highlight this point better, in Fig. 10, we averaged
values over the energy bins.

Figure 10: Charge excess to geomagnetic ratio on average for simulations with
proton or iron primaries and various energies

To measure the impact of the geomagnetic angle on our results, we represent in
Fig. 11 the relative deviation between $\sin{\alpha}$ at the GRAND site with
magnetic inclination $I_{B}=60.79$°(left) and at the Auger site with magnetic
inclination $I_{B}=35.2$°(right). We find that although the Auger site is a
location with particularly high Askaryan contribution to the radio-emission,
the relative deviation of $\sin{\alpha}$ between the GRAND and the Auger site
is only up to few tens of percent for very inclined showers. Significant
deviations are observed only for showers with azimuth angle towards North and
with low zenith angle (close to 60°). Between different locations on Earth,
there are also variations of the magnetic field strength that can be roughly
up to a factor 2. This implies that for very inclined showers
($\theta>80\degree$) the combined effect of the geomagnetic angle and the
magnetic field strength can lead up to a factor 4 between the Askaryan ratio
at a favorable site for the geomagnetic emission (such as the GRAND site) and
a non favorable site (Auger site). Still, as we can see in Figure 9, for the
GRAND site at 80°, the Askaryan ratio is of $\sim 2\%$ which implies that even
with an additional factor 4 we expect a dominant geomagnetic emission. This
justifies the assumption of a radio-emission dominated by the geomagnetic
mechanism for inclined showers, independently of the location on Earth.

Figure 11:  Relative difference of $\sin{\alpha}$ ($\alpha$ being the
geomagnetic angle): $(\sin{\alpha_{\rm Dunhuang}-\sin{\alpha_{\rm
Auger})/\sin{\alpha_{\rm Dunhuang}}}}$ as a function of the shower azimuth and
zenith angle between the Dunhuang site with magnetic field inclination
$I_{B}=60.79$°and the Auger site with $I_{B}=35.2$°. $\Phi=0$ corresponds to a
shower towards North and the azimuth angle is counted positively towards Est.

###  5.3 Effects of signal processing and trigger conditions

We now perform a similar study for realistic time traces, i.e., applying to
the traces the full treatment described in Section 3 and then a trigger
condition at the antenna level on the reconstructed amplitude $\sqrt{I}$ of
either $E_{\rm trig,3\sigma}=60\,\mu$V/m (3$\sigma$ trigger threshold, left
panel) or $E_{\rm trig,5\sigma}=100\,\mu$V/m (5$\sigma$ trigger threshold,
right panel). In the following, we will always present results for 3 distinct
cases: no processing (i.e. raw traces), 3$\sigma$ trigger threshold
($\sigma_{\rm noise}=20\mu$V/m, $E_{\rm trig,3\sigma}=60\mu$V/m + filtering
and sampling) and 5$\sigma$ trigger threshold ($\sigma_{\rm noise}=20\mu$V/m,
$E_{\rm trig,5\sigma}=100\mu$V/m + filtering and sampling).

Figure 12: Charge excess to geomagnetic ratio as a function of the energy and
the zenith angle for simulations with a proton primary on a star-shape layout,
considering traces with noise and a $3\sigma$ trigger threshold (left) or a
$5\sigma$ trigger threshold (right). Each data point corresponds to the
average ratio for simulations with the same energy and zenith angle and the
error bars to the RMS of each distribution.

In Figure 12, we represent the $a$-ratio derived from Eq. 8, 9 and 10 using
this time-processed outputs. Note that this $a$-ratio does not directly
correspond to the physical ratio between the charge excess and geomagnetic
emission, as presented in the previous section, but rather to the expected
measurement in an experimental case. We observe for showers with low
inclination a similar behavior as in Fig. 9. However, for most inclined
showers ($\theta>70\degree$), we can also observe that above a certain zenith
angle the ratio increases with inclination, in particular at low energy. This
can be understood considering that, the more inclined the shower, the more
diluted the radio signal, because the shower develops over longer distances
[1]. As a consequence, for the most inclined showers we will have an
increasing number of antennas with low amplitude signals (mostly for low
energy showers). This implies that for such showers the amplitude of the
charge excess mechanism is comparable to the noise amplitude, inducing a bias
in the $a$ ratio.

This plot also allows us to infer an interesting feature: for cosmic-ray
inclined air-showers (with $\theta>65^{\circ}$), the charge excess to
geomagnetic ratio drops below 5% independently of the trigger threshold for
energies above 0.2 EeV. As the geomagnetic emission is along ${\bf
v}\times{\bf B}$, it is perpendicular to the local direction of the magnetic
field. Consequently, in the case of a very dominant geomagnetic emission,
which is expected for inclined air-showers, the total electric field should
also be in good approximation perpendicular to the direction of the magnetic
field. This is a specific feature that could enable to discriminate between
the radio signal from cosmic-ray air-showers and the background noise,
providing an efficient way to perform autonomous radio detection.

In the following sections we quantify more precisely this effect for inclined
air-showers induced by cosmic-rays and establish a criterium for a trigger at
the DAQ level.

##  6 Shower identification method

In the previous section, we highlighted that radio emission from inclined EAS
is strongly polarised in a direction orthogonal to $\mathbf{B}$, a specific
feature which could thus be used to allow for autonomous radio-detection, i.e.
a detection of air-showers using the radio signal only for antennas with 3
polarisation channels. In this section, similarly to what was presented in
[23] we define the trigger thresholds required to perform such an
identification, and apply our method to a realistic array layout.

###  6.1 Threshold definitions for a trigger at the DAQ level

####  6.1.1 Projection of the electric field along $\mathbf{B}$

In this section, we focus only on showers with a zenith angle between
80$\degree$ and 90$\degree$. We consider a set of $\sim 11\,000$ simulations
for proton and iron-induced showers, with primary particle energies, azimuth
and zenith angles given in Table 2. We aim at establishing a criterium to
trigger through an online treatment at the DAQ level with a strong background
rejection. For this purpose, we focus on the fraction of the total electric
field along the direction of the magnetic field, i.e., $E_{b}/E_{\rm tot}$,
where $E_{\rm tot}$ is the total amplitude of the electric field, ${E_{\rm
tot}}=\sqrt{E_{x}^{2}+E_{y}^{2}+E_{z}^{2}}$ and $E_{b}$ is the absolute value
of the projection of the total electric field along the direction of ${\bf
B}$, $E_{b}=|\mathbf{E}_{\rm tot}\cdot\mathbf{u}_{b}$| with $\mathbf{u_{b}}$
the unit vector along the direction of the magnetic field. As the geomagnetic
emission is perpendicular to ${\bf B}$, the only contribution to $E_{b}$ comes
from the charge excess and for a dominant geomagnetic emission $E_{\rm tot}$
is similar to $E_{\rm geo}$. Therefore, the ratio $E_{b}/E_{\rm tot}$ should
be quite similar to the $a$ ratio. However using this ratio has 2 main
advantages:

  * 1.

First, it can be computed at the DAQ level directly after a preliminary
reconstruction of the direction of origin of the radiation. a necessary step
to deconvolve the antenna response from the voltage signal and thus provide a
fast estimate of the electric field amplitude, while for the $a$-ratio,
calculations have to be performed using the amplitude of the electric field in
the shower plane, which requires information about the shower geometry. The
reconstruction of the shower direction for very inclined air-showers has been
studied in [34] and shows that a plane angular reconstruction at the DAQ level
of 1° should be achievable from the raw timing information. The impact of
misreconstructed events in our background rejection efficiency is however not
considered in this paper and should be quantified in further studies.

  * 2.

Second, as $E_{b}$ corresponds to a projection of the charge excess, we have
$E_{b}$<$E_{\rm ce}$ and since $E_{\rm geo}<E_{\rm tot}$, $E_{b}/E_{\rm tot}$
allows to put even more restrictive constraints on the trigger condition.

In Fig. 13, we represent histograms of the $E_{b}/E_{\rm tot}$ ratio at the
antenna level for our set of $11\,000$ proton and iron simulations in the 3
following cases: no processing, 3$\sigma$ trigger threshold and 5$\sigma$
trigger threshold (left panel) and comparing proton and iron primaries at the
5$\sigma$ trigger level (right panel). Note that for histograms at the
$3\sigma$ and $5\sigma$ trigger thresholds, the effective number of antennas
is reduced as some are cut with the trigger condition. From the plot on the
left panel we attest, as could already be inferred in Fig. 12, that the ratio
is distributed towards higher values in the 3$\sigma$ and 5$\sigma$ case than
in the no processing case because of the noise added. As expected, we also
observe a slightly larger number of antennas with a $3\sigma$ threshold than a
$5\sigma$ threshold, as we considered a more aggressive trigger condition.
Considering now the right panel of Fig. 13, very similar results for proton
and iron primaries can be found, in agreement with Fig. 10. The most
interesting feature is that independently of the primary or the trigger
threshold considered, the expected $E_{b}/E_{\rm tot}$ ratio drops to very low
values for the large majority of antennas: it is below 7% for 86% (93%) of
antennas at the $3\sigma$ ($5\sigma$) trigger threshold. This is a strong
signature for air-showers which can be used at the DAQ level.

Figure 13:  Histogram of the $E_{b}/E_{\rm tot}$ ratio measured at each
antenna for a set of about $11\,000$ simulations for (left) proton induced
showers with different trigger thresholds and (right) proton or iron induced
showers and a $5\sigma$ trigger threshold.

In Fig. 14, we represent the distribution of the $E_{b}/E_{\rm tot}$ ratio as
a function of the amplitude $E_{\rm tot}$ on a 2D histogram for the set of
$11\,000$ simulations, at the $5\sigma$ trigger threshold (left panel). On the
right panel, we average the values of the ratio for different bins of the
amplitude with either a 3$\sigma$ or 5$\sigma$ trigger threshold (right
panel).

Figure 14: (Left) $E_{b}/E_{\rm tot}$ ratio as function of the total amplitude
$E_{\rm tot}$ measured by each antenna on a star-shape layout for a set of
$\approx\,11\,000$ simulations on a star-shape layout considering a $5\sigma$
trigger threshold. (Right) $E_{b}/E_{\rm tot}$ ratio as function of the total
amplitude measured by each antenna and averaged for different bins of the
total amplitude, considering either a $3\sigma$ or a $5\sigma$ trigger
threshold.

It appears that the highest values of the $E_{b}/E_{\rm tot}$ ratio are
related to antennas with low amplitude. This can be understood as for these
antennas the charge excess amplitude is subdominant comparatively to the noise
so that our measurement of the $a$ ratio ratio becomes closer to the
noise/geomagnetic ratio. Hence, to strengthen our trigger criteria we could
define a trigger condition that depends on the total amplitude measured by
each antenna. We could for example define a trigger condition $E_{b}/E_{\rm
tot}<f(E_{\rm tot})$ with $f(E_{\rm tot})$ obtained by fitting the points
$\langle E_{b}/E_{\rm tot}\rangle+n\sigma_{E_{b}/E_{\rm tot}}$, $n$ being an
integer and $\sigma$ corresponding here to the RMS of the distribution.

In Fig. 14, we fit $f(E_{\rm tot})$ with the following trigger function

$f(E_{\rm tot})=a+\frac{b}{E_{\rm tot}}$ (16)

For $n=1$, we find $a=0.1620\pm 0.0005$ and $b=6\pm 0.1$ which results in a
$83\%$ trigger efficiency and for $n=3$ we have $a=0.0340\pm 0.0009$ and
$b=11\pm 0.3$ resulting in a $99\%$ trigger efficiency. As modeling the
transient noise is challenging, it is not possible to quantify directly the
fraction of background events that will be rejected. However, selecting an
$E_{b}/E_{\rm tot}$ ratio $\leq 7$% restricts our field of view to only $7\%$
of the full solid angle as illustrated in Fig. 15). Hence, assuming an
isotropic polarisation for the noise orientation this cut should reduce the
number of noise triggered events by a factor $\sim 15$ while more than 93% of
antennas pass this cut for cosmic-ray signals for a 5$\sigma$ threshold. This
provides a strong noise rejection efficiency, even without knowing any of its
features. To quantify the impact of the geographic location on the
$E_{b}/E_{\rm tot}$ ratio, we can also compute our background rejection
efficiency in the least favorable case. Following the text at the end of
Section 5.2, for a non favorable site, the Askaryan ratio should scale by a
factor 4 compared to the one computed at the GRAND site. Hence, the
$E_{b}/E_{\rm tot}$ ratio should also roughly scale by a factor 4 which lead
to a new value of $(E_{b}/E_{\rm tot})_{\rm non\,favorable}=0.07\times
4=0.28$. With this new value we find a background rejection efficiency of
$72\%$.

Yet the selection treatment presented here requires that the electric field
amplitude is reconstructed by deconvolving the output voltage from the antenna
response. As it is strongly anistropic, this treatment requires that the
direction of origin of the signal is known, which is therefore possible only
at the central DAQ level, by combining the timing information of different
triggered detection units. Interestingly enough, the raw voltage information
available at the individual antenna level already provides some information on
the nature of the signal and should allow for a first rough —but efficient—
rejection of background signals. To illustrate this, we represent in Fig. 16
the quantity $V_{b}/V_{\rm tot}$ (i.e., the voltage ratio computed in a same
way as the $b$-ratio $E_{b}/E_{\rm tot}$ ) with a $5\sigma$ trigger threshold.
Even though the orthogonality to the magnetic field vector is not as strong as
for the electric field, we find that the voltage $b$ ratio is below $28\%$ for
$90\%$ of antennas. Hence even though the limits on the voltage are not as
stringent as the electric field, these results show that a trigger directly at
the antenna level (i.e. at the first level trigger) could already be performed
and reject 72% of noise events.

Figure 15: Sketch of the electric field vector $E_{\rm tot}$ and its
projection on the magnetic field of the Earth, $E_{b}$. With a fixed cut on
the $E_{b}/E_{\rm tot}$ ratio, the field of view is restricted to the green
area and all the red area corresponds to rejected events.

####  6.1.2 Polarized intensity

All the previous results concerning the identification of shower signatures
were based on one unique observable, the $E_{b}/E_{\rm tot}$ ratio. The limits
defined on this observable in the previous section seem already strong enough
to perform an efficient background rejection. However, one may want to explore
the possibility to highlight other criteria for an identification of shower
signatures, as the combination of all the criteria would allow to put even
more restrictive constraints on the trigger condition ensuring an even more
robust autonomous detection. Here, we explore the degree of polarisation of
the radio signal from air-showers. It has been well established that such
signals should be highly polarised [16], thus we aim at quantifying this
feature still in the perspective of a trigger at the DAQ level.

Figure 16:  Comparison between of the “$b$-ratio" for the electric field:
$E_{b}/E_{\rm tot}$ and for the voltage: $V_{b}/V_{\rm tot}$.

The calculation of Stokes parameters as defined in Section 4.1 requires only
the components of the electric field. Then, instead of computing these
parameters in the shower plane, we could also directly compute them at the DAQ
level, using the 3 components in the geographic plane $E_{x}$, $E_{y}$,
$E_{z}$. We can thus define the degree of polarisation of the signal as
$I_{p}/I$, where $I$ corresponds to the Stokes parameter related to the total
intensity of the signal and $I_{p}=\sqrt{Q^{2}+U^{2}+V^{2}}$ corresponds to
the polarised intensity.

Figure 17:  Histogram of $1-I_{p}/I$ (where $I_{p}/I$ is the fraction of
polarised intensity) measured at each antenna for proton induced showers on a
star-shape layout at either a $3\sigma$ or a $5\sigma$ trigger threshold.

The distribution of $1-I_{p}/I$ is represented in Fig. 17 for the star-shape
simulation set. We retrieve a highly polarised signal with more than $98\%$ of
antennas having $1-I_{p}/I$ below $0.02$ at the $5\sigma$ trigger threshold.
This specific feature could also provide an efficient way to strengthen our
criteria for autonomous radio detection as we could set very low limits on
$1-I_{p}/I$ without cutting the signal from any air-showers. It should however
be noted that most of noises such as humans emissions are also expected to be
highly polarised, hence this observable may not be as efficient as it seems.
Still, it could be used as backup to reinforce the criterion defined in the
previous section. Setting the limits of the $E_{b}/E_{\rm tot}$ ratio to 7%,
we already expect to cut $93\%$ of the noise, then among the 7% remaining, the
criterion over $I_{p}/I$ could help us evince all that are not strongly
polarised, guaranteeing an even more efficient detection of air-showers.

###  6.2 Application to a realistic layout

All the results in the previous section were presented for a star-shape layout
which consists of a plane layout with a very dense array of antennas and for
which the air-shower core is always centered on the layout center. In this
section, we apply the same treatment to an experimental layout. The HotSpot 1
layout (hereafter HS1), is a rectangular layout made of $10\,000$ antennas
[13], each with a spacing of 1 kilometer as illustrated in Fig. 18. It
corresponds to a real geographic location (centered at latitude
$=42.1\degree$, longitude $=86.3\degree$) in the XinJiang province in China,
used to evaluate the potential of GRAND for neutrino detection in [13].

Figure 18: Topography of the Hotspot 1 layout from [13], corresponding to a
giant rectangular array of $10\,000$ antennas over $10\,000$ square kilometers
with simulated location in Ulastai (China).

Here, we consider a set of about $1\,000$ simulations of proton or iron
induced showers corresponding to 9 energy bins, 7 zenith angles and 4 azimuth
angles (see Table 3). For each simulation, the core position is generated
randomly, resulting in a varying number of antennas inside the radio
footprint.

####  6.2.1 Projection of the electric field along B

In Fig. 19, we present histograms of the $E_{b}/E_{\rm tot}$ ratio for proton
and iron induced showers at the $5\sigma$ trigger threshold, similarly as in
Fig. 13. As previously observed, we retrieve for the HS1 layout a strong
similarity between proton and iron induced showers, suggesting that for
inclined showers, the differences regarding the density at $X_{\rm max}$ for
both of these primaries do not impact significantly the charge excess to
geomagnetic ratio. Due to this high similarity, all the following results in
this section will be presented for protons only. We also note that there is a
larger number of antennas that trigger for proton than for iron simulations.
This is related to the fact that the $X_{\rm max}$ position for iron-induced
showers occurs higher in the atmosphere than for proton showers resulting in a
larger footprint and a more diluted signal. Figure 19 also shows that even for
an experimental layout such as HS1, we retrieve low values for the
$E_{b}/E_{\rm tot}$ ratio, with more than 87% of antennas with a ratio below
7% at the $3\sigma$ trigger threshold level and about 93% at the $5\sigma$
trigger threshold level.

Figure 19:  Histogram of the $E_{b}/E_{\rm tot}$ ratio measured at each
antenna for proton and iron induced showers at the $5\sigma$ trigger threshold
on the HS1 layout.

In Fig. 20, we represent histograms of the $E_{b}/E_{\rm tot}$ ratio for
proton induced showers at the $3\sigma$ trigger level comparing results from
the star-shape and the HS1 layout. This reveals a particularly interesting
feature: although the HS1 and the star-shape consist of very different layout,
we retrieve a high similitude between their $E_{b}/E_{\rm tot}$ ratio
distributions. Such similitude can be justified considering that due to the
trigger condition on the amplitude, the antennas that trigger are likely to be
the closest antennas to the Cerenkov cone independently of the layout
configuration. This suggests that our criterion defined to perform autonomous
radio detection seems to be almost independent of the considered layout and
could therefore be implemented for a wide variety of experiments dedicated to
the radio detection of air-showers.

Figure 20:  Normalised histogram of the $E_{b}/E_{\rm tot}$ ratio measured at
each antenna for proton induced showers at the $3\sigma$ trigger threshold for
either the HS1 either the star-shape layout.

Finally in Fig. 21 we represent the $E_{b}/E_{\rm tot}$ ratio as a function of
the amplitude in a 2D plot at the $3\sigma$ trigger threshold (left panel) and
after averaging the ratio over different bins of the total amplitude for both
$3\sigma$ and $5\sigma$ trigger thresholds (right panel). Here again, we
retrieve similar results with the ones of the star-shape layout, that is that
the $E_{b}/E_{\rm tot}$ ratio decreases with increasing amplitude of the
signal measured by the antennas, even though here our statistic is a bit
reduced as seen on the left panel. We can once again define a trigger
condition $E_{b}/E_{\rm tot}<f(E_{\rm tot})$ where $f(E_{\rm tot})$ is
obtained by fitting the points $\langle E_{b}/E_{\rm
tot}\rangle+n\sigma_{E_{b}/E_{\rm tot}}$ with Eq. 16. For $n=1$, we find
$a=0.1775\pm 0.0004$ and $b=6\pm 0.1$ with a $83\%$ trigger efficiency and for
$n=3$ $a=0.0330\pm 0.0007$ and $b=10.9\pm 0.3$ with a $99\%$ trigger
efficiency. This is in excellent agreement with the results of the star-shape
simulation.

Figure 21: $E_{b}/E_{\rm tot}$ ratio measured at each antenna for proton
induced showers at the $3\sigma$ trigger threshold on the HS1 layout (left)
and after averaging the ratio over different bins of the total amplitude for
both $3\sigma$ and $5\sigma$ trigger thresholds (right).

####  6.2.2 Polarized intensity

In Fig. 22, we represent an histogram of $1-I_{p}/I$ computed at each antenna
for simulations of proton induced showers on the HS1 layout for both trigger
thresholds $3\sigma$ and $5\sigma$. It appears that the ratio is below 2% for
more than 90% of the antennas at the $3\sigma$ trigger threshold and more than
98% at the $5\sigma$ trigger threshold, which confirms that the signal from
air-showers is highly polarized.

Figure 22:  Histogram of $1-I_{p}/I$ (where $I_{p}/I$ is the fraction of
polarised intensity) measured at each antenna for proton induced showers on
the HS1 layout at either a $3\sigma$ or a $5\sigma$ trigger threshold.

##  7 Polarisation signatures of neutrino-induced air-showers

Ultra-high energy neutrinos can also induce extensive air-showers. As these
particles have a very low cross section, they need a dense environment to have
a reasonable probability to interact. This can happen for example in the case
of an up-going shower, where the neutrino interacts directly in the ground and
decays into a tau particle. The tau particle can then emerge and decay in the
atmosphere inducing an extensive air-shower. Antennas placed in a valley
provide an alternative. A down-going tau neutrino can in this case interact
within a mountain, produce a tau particle, which in turn escapes from the
mountain and decays in the atmosphere, inducing an extensive air-shower as
illustrated in Fig. 16 of [13].

Only Earth-skimming trajectories correspond to non-negligeable detection
probabilities. Neutrino showers are therefore expected to develop close to the
ground, in denser environment than cosmic-ray showers. This signature should
directly translate into a higher $a$-ratio. Indeed, we recall as can be seen
from Eq. (15), that a higher density should result in an increase of the
charge excess (increase of $C_{x}$) and a decrease of the geomagnetic emission
(decrease of $\langle v_{d}\rangle$).

In this section, we study this trend for neutrino induced air-showers by
comparing their $E_{b}/E_{\rm tot}$ ratios with the ones we obtained for
cosmic-ray showers. For this purpose, we consider $1\,000$ simulations of
neutrino showers, either down-going or up-going with zenith angles ranging
from 85$\degree$ to 95$\degree$ ($\theta>90\degree$ for up-going showers) and
various energies ranging from $0.1\,$EeV to $25\,$EeV. Simulation were
performed on the HS1 layout as presented in Fig. 18 with different values for
energy and direction of origin.

In Fig. 25, we represent the $E_{b}/E_{\rm tot}$ ratio for neutrino showers
comparing a conservative and aggressive trigger threshold (left panel) and
comparing neutrinos, proton and iron nuclei at the $5\sigma$ trigger threshold
level (right panel). It appears clearly that the ratio is higher for neutrino
than for cosmic-ray induced showers: the distribution peaks around 15% for
neutrino primaries and 2% for cosmic-ray showers. Still, we can also observe
that for some antennas, even in the case of neutrino simulations, the
measurement of the $E_{b}/E_{\rm tot}$ ratio can reach low values, close to 0.
This is related to the radial signature of the charge excess, implying that
its emission should drop to 0 for antennas close to the core position or for
antennas along the ${\bf v\times(v\times B)}$ baseline, as the charge excess
and geomagnetic emission are both orthogonal to $B$ along this axis. This can
also be seen from Eq. 15: for showers occurring in denser environments, the
slope of the $a$-ratio will be steeper due to an enhancement of $C_{x}/\langle
v_{d}\rangle$, but the ratio still tends to 0 for $\omega=0$.

In Fig. 23, we also represent the $E_{b}/E_{\rm tot}$ distribution for
neutrino showers, but this time differentiating between up-going and down-
going showers. It appears that the $E_{b}/E_{\rm tot}$ ratio is higher for up-
going than down-going showers. This trend can be justified by geometrical
considerations as illustrated in Fig. 24. The $\omega$ angle refers to the
angular deviation between the shower axis and the direction that goes from
$X_{\rm max}$ to a given antenna. The $\omega$ angle is counted positively or
negatively following the trigonometric direction. On our sketch, it can be
seen that for $\omega>0$, the $X_{\rm max}$-antenna distance increases with
$|\omega|$ and that for $\omega<0$, the $X_{\rm max}$-antenna distance
decreases as $|\omega|$ increases. Yet, for up-going showers, we mostly detect
antennas with $\omega<0$, i.e., antennas that are closer to the emission point
as $|\omega|$ increases. This bias towards antennas with $\omega<0$ for up-
going showers implies that antennas with high $|\omega|$ values receive a less
diluted signal and are more likely to trigger. This results in a higher
$E_{b}/E_{\rm tot}$ ratio as was observed in Fig. 8.

Figure 23:  $E_{b}/E_{\rm tot}$ distribution comparing neutrino and cosmic-ray
primaries on the Hotspot 1 layout at a $3\sigma$ trigger threshold. Figure 24:
Sketch of angular distance $\omega$ for down-going (left) and up-going (right)
showers. Late antennas are more frequent for down-going than up-going showers,
also the $X_{\rm max}$-antenna distance increases with the $\omega$ angle for
late-antennas although it decreases with the $\omega$ angle for early-
antennas.

Figure 25: Histograms of the $E_{b}/E_{\rm tot}$ ratio for neutrino
simulations on the Hotspot 1 layout at different trigger thresholds (left) and
comparing different primaries at a $5\sigma$ trigger threshold(right).

Finally in Fig. 26, we represent the comparison between the $E_{b}/E_{\rm
tot}$ ratio as a function of $\omega$ for neutrino and cosmic-ray induced
showers. It appears clearly that for a given $\omega$, values of the ratio
will be higher for neutrinos than for cosmic-ray showers. As a consequence, we
can infer that differences in the values of the ratio for neutrino and cosmic-
ray showers as observed in Fig. 25 are not (or at least not only) due to a
different distribution of the antennas but density effects do play a role.

Figure 26: $E_{b}/E_{\rm tot}$ ratio as a function of $\omega$ for proton,
iron and neutrino induced showers at a $5\sigma$ trigger threshold, on the
Hotspot 1 layout.

These results have several implications

  1. 1.

Although the criteria defined in the previous section to perform autonomous
radio detection are very efficient for cosmic-ray induced showers, it would
have a much reduced efficiency for neutrino induced showers if applied as
such. For experiments dedicated to the detection of neutrinos and cosmic-ray
showers, the criteria defined in Section 6.1 can be implemented only if a
preliminary discrimination between neutrino and cosmic-ray showers can also be
performed directly at the DAQ level.

  2. 2.

It could be possible to discriminate between cosmic-ray and neutrinos induced
showers using the $E_{b}/E_{\rm tot}$ ratio, as the ratio is expected to be
distributed towards higher values for neutrinos induced showers. However, due
to the radial signature of the charge excess, the $E_{b}/E_{\rm tot}$ ratio
will drop to 0 for antennas close to the core or on the ${\bf v\times(v\times
B)}$ axis, implying that the discrimination between neutrino and cosmic-ray
showers could not be performed at the antenna level for all the antennas but
rather offline, averaging for example the ratio over the full array.

  3. 3.

Finally, as the contribution of the charge excess emission is stronger for
neutrino showers, it might be possible to use this charge excess signature to
reconstruct air-shower parameters. For example, as illustrated in Fig. 1, the
charge excess emission points towards the air-shower core position and as
shown in Fig. 8, the charge excess amplitude increases with the distance to
the core. These are two strong signatures that could help us reconstruct the
air-shower core position, allowing for angular reconstruction if the $X_{max}$
position can also be reconstructed, for example. This will be investigated in
further dedicated studies.

##  8 Conclusion

Based on numerical simulations of inclined cosmic-ray and neutrino air-
showers, we have characterized polarisation signatures of the radio signal,
that could serve to perform autonomous radio detection at the DAQ level.

We focused on the two main contributions to the polarisation, namely the
geomagnetic and the charge excess emissions. We have evidenced the clear
predominance of the geomagnetic emission over the charge excess emission for
inclined ($\theta>65\degree$) cosmic-ray air-showers. This implies that the
total electric field from the radio emission of such air-showers should be in
good approximation perpendicular to the direction of the local magnetic field.

This specific feature can be exploited to reject noise and identify cosmic-ray
triggered events. We have established trigger conditions directly at the DAQ
level, based on the fraction of the total electric field projected along
$\mathbf{B}$ ($E_{b}/E_{\rm tot}$). This quantity can be computed after
reconstructing the electric field by deconvoluting the voltage from the
antenna response, and is by construction sensitive to the geomagnetic emission
polarisation. This of course requires first that a reconstruction of the
shower direction is performed.

We tested our rejection method on both a star-shape and an experimental
layout. For the experimental layout in particular, our results show that a
uniform trigger condition on the $E_{b}/E_{\rm tot}$ ratio at 7% enables a
rejection efficiency of $\approx 93\%$ for a favorable site (as defined in
section 5.2), and $\approx 72\%$ for a non-favorable one. The trigger
efficiency of this cut for cosmic-ray air-showers is $86\%$ (93%) with a
$3\sigma$ ($5\sigma$)-trigger threshold level on the total amplitude. Note
that these values were obtained assuming an isotropic distribution in space
and a gaussian distribution in amplitude of noise events and that the real
noise environment of the target site should be used to get more accurate
estimations. Moreover, we show that the accuracy of our trigger criteria for
shower identification increases with the amplitude of the electric field.
Hence implementing an amplitude-dependent trigger condition would strengthen
our background rejection efficiency. As we expect the signal from air-showers
to be highly polarised, the fraction on polarised intensity $I_{p}/I$ could
also help perform shower identification.

On the other hand, for neutrino-induced showers, we find that the higher
contribution of the charge excess emission does not enable to perform shower
identification using the same method as for cosmic-rays. However, the charge
excess polarisation pattern should enable to perform an efficient
reconstruction of the air-shower parameters as well as a discrimination
between cosmic-ray and neutrino induced showers.

The method developed here is largely independent of the detector array
configuration, which suggests that it could be applied to a large variety of
experiments with high efficiency.

### Acknowledgments

We thank Valentin Decoene, Eric Hivon, and Simon Prunet for very fruitful
discussions. We are also grateful to Harm Schoorlemmer for his valuable input
at the beginning of this project. This work is supported by the APACHE ANR
grant (ANR-16-CE31- 0001) of the French Agence Nationale de la Recherche, the
Programme National des Hautes Energies of CNRS/INSU with INP and IN2P3, co-
funded by CEA and CNES, and the European Research Council under the European
Unions Horizon 2020 research and innovation programme (grant agreement No.
805486). Simulations were performed using the computing resources at the CC-
IN2P3 Computing Centre (Lyon/Villeurbanne – France), partnership between
CNRS/IN2P3 and CEA/DSM/Irfu.

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