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

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 $(1/\rho)^J$, where $\rho$ 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.