Questioning the relationship between the $\chi$4 susceptibility and the dynamical correlation length in a glass former

Clusters of fast and slow correlated particles, identified as dynamical heterogeneities (DHs), constitute a central aspect of glassy dynamics. A key factor of the glass transition scenario is a significant increase of the cluster size xi(4) as the transition is approached. In need of easy-to-compute tools to measure xi(4), the dynamical susceptibility chi(4) was introduced recently, and used in various experimental studies to probe DHs. Here, we investigate DHs in dense microgel suspensions using image correlation analysis, and compute both chi(4) and the four-point correlation function G(4). The spatial decrease of G(4) provides a direct access to xi(4), which is found to grow significantly with increasing volume fraction. However, this increase is not captured by chi(4). We show that the assumptions that validate the connection between chi(4) and xi(4) are not fulfilled in our experiments.