Revisiting the Radio Lateral Distribution Function: An amplitude dependence on $X_{\rm max}$ and primary composition

TL;DR

This work shows a strong $X_{ m max}$ dependence of the ground-level radio LDF amplitude, explained by two competing scalings $(1/R)$ and $(1/ ho)^{J( heta)}$ with coherence loss $J( heta)$. Using ZHAireS simulations at GRAND and Auger, the authors validate how zenith angle and geomagnetic field modulate which scaling dominates, leading to a composition-sensitive amplitude signal that persists even when EM energy is normalized. The findings imply that primary mass information is encoded in both the LDF shape and its brightness, enabling potential event-by-event composition inference without explicit $X_{ m max}$ reconstruction, while also highlighting possible biases in EM energy reconstruction methods that ignore amplitude dependencies. Overall, the paper provides a cohesive physical picture connecting $X_{ m max}$, coherence, and ground-field amplitudes, with practical implications for radio-based cosmic-ray analyses.

Abstract

We show that there is a strong dependence of the radio LDF electric field amplitudes at ground level on the position of $X_{\rm max}$ in the atmosphere, even accounting for differences in the EM energy of the showers. Since an $X_{\rm max}$ dependence leads to a primary composition dependence, this implies that information on the mass composition is encoded not only in the LDF shape but also in its amplitude. This $X_{\rm max}$ dependence can be explained in terms of two competing scalings of the measured electric field: One goes with $(1/ρ)^J$, where $ρ$ is the air density at $X_{\rm max}$ and $J$ is a zenith dependent non-linearity factor describing coherence loss. This density scaling tends to decrease the geomagnetic emission of deeper showers. The other scaling goes with $(1/R)$, where $R$ is the distance from $X_{\rm max}$ to the core at ground, and instead increases the measured electric field of deeper showers. At low zenith angles, the $(1/R)$ scaling is stronger and leads to larger measured electric fields as $X_{\rm max}$ increases. The picture at higher zeniths, i.e., lower densities, is more nuanced. In this region, the deflections due to the Lorentz force are much larger and introduce extra time delays between the particle tracks, decreasing the coherence of the emission. This loss of coherence is highly dependent on the strength of the geomagnetic field and can slow down, or even reverse the increase of the radio emission with decreasing air density. This strong, yet historically overlooked LDF amplitude dependence on $X_{\rm max}$/composition could be used to directly infer, even bypassing any $X_{\rm max}$ reconstruction, the cosmic ray primary composition on an event-by-event basis. It could also have some repercussions on other radio reconstruction methods, such as a possible $X_{\rm max}$/composition bias on shower electromagnetic energy reconstruction methods.

Figures (7)

• Figure 1: Left: Random forest normalized feature importances for events with zenith angle $\theta=62^\circ$ at the Giant Radio Array for Neutrino Detection (GRAND) site. For each triggered antenna in an event, there are two features: the distance to the axis (blue) and the peak amplitude (green). The antennas are ordered in increasing distance to the axis (from left to right). Right: LDFs obtained from full ZHAireS simulations of events coming from the North with $\theta=62^\circ$ at GRAND, normalized by the EM energy of each shower. These simulations were the ones used as input for RDSim in order to generate the training and testing events for the random forest. Note that RDSim morphs these input simulations to generate events with different azimuth angles and core positions. The vertical dashed lines, starting from the left, represent the average distance, for all events, to the closest antenna (first line), second closest antenna (second line) and so on. See text for more details.
• Figure 2: LDFs obtained from full ZHAireS simulations of events coming from the North with $\theta=62^\circ$ at GRAND. Left: Original LDF, without EM energy normalization (all showers have the same primary energy $E_0=1.25$ EeV). Proton-induced showers are shown in red and iron in blue. Right: Same as left, but the electric fields were normalized by the EM energy of each shower (all showers now have the same EM energy $E_{EM}=1.25$ EeV).
• Figure 3: Radio LDFs from full ZHAireS simulations for several zenith angles at the GRAND site (top) and AUGER site (bottom). All LDFs were normalized by the EM energy of each shower.
• Figure 4: Values of $R$ (red), $\rho$ (blue) and $\rho^{J(\theta)}$ (black) as a function of $X_{\rm max}\,$ for several zenith angles and experimental sites, normalized by their respective minimum values for each zenith angle. $J(\theta)$ is a non-linearity factor used to take into account coherence loss (see section \ref{['sec:lossofcoherence']}). Top left: Auger site at $\theta=55^\circ$. Top right: Auger site at $\theta=80^\circ$. Bottom left: GRAND site at $\theta=42^\circ$. Bottom right: GRAND site at $\theta=82^\circ$. Note that, except for the fitted $J(\theta)$ values (see section \ref{['sec:lossofcoherence']}), these plots did not use the results of the simulations and instead were constructed solely using the shower geometry (zenith angle, ground altitude and $X_{\rm max}\,$) along with a model of the atmosphere (U.S. Standard Atmosphere 1976 USatm76).
• Figure 5: Top left: Estimate of the Askaryan and geomagnetic emission at GRAND and Auger, obtained from the complete set of full ZHAireS simulations. In this panel, the values were normalized by the Askaryan amplitude. Top right: Same as top left, but normalized by the geomagnetic emission. Bottom left: Askaryan and geomagnetic amplitudes after applying the inverse distance and density scalings, normalized by the scaled geomagnetic amplitude. Bottom right: Non-linearity factor $J(\theta)$, which represents the coherence loss, fitted to the simulations at the GRAND site and for each zenith angle, separately. For more details on all panels, see the text.